ON THE GEOMETRIES OF NON-ORIENTABLE

FIBRE-BUNDLES WITH BASE AND

FIBRE EUCLIDEAN SURFACES

Robin J. Cobb

Abstract. We show that closed 4-manifolds which fibre over the torus or Klein bottle with fibre a torus or Klein bottle are geometric in the sense of Thurston.

Introduction. The purpose of this paper is to show that those 4-manifolds M that are total-spaces of F -bundles over S, where F and S are closed surfaces of genus 1, are geometric in the sense of Thurston. Ue [Ue1] has shown this to be true in the instances where M is orientable. Below we complete the above task by showing that those non-orientable bundle-spaces are also geometric. We provide a listing of all such manifolds M up to bundle-diffeomorphism type, and display an explicit embedding of the fundamental group Π_{1}(M) into the isometry group of some standard model, endowing M with a geometric structure.

This is achieved by primarily noting [Oh] that Π_{1}(M) largely determines the diffeomorphism type of such 4-manifolds. After generating a listing of group presentations of all possible Π_{1}(M), the determination of bundle-diffeomorphism classes of M is reduced to that of group extension isomorphism classes of the associated short exact sequences of the given fibrations. Thenceforth we provide a faithful and discrete representation Φ of Π_{1}(M) into the isometry group of one of the 4dimensional geometries X =: E^{4}, Nil^{3} × E, Sol^{3} × E or Nil^{4}; the quotient of X by the image of Φ yields a compact manifold diffeomorphic to M.

Nomenclature. Throughout, the symbols F and S denote closed surfaces of genus 1. Here we take the fundamental groups of such surfaces to be each generated by an ordered pair of loops Π_{1}(F) = ⟨γ, δ⟩, Π_{1}(S) = ⟨α, β⟩ together with the single relator:

≀α, β≀_{S} = \{ \begin{matrix}[α, β], & if S is the torus T^{2}; \\ αβα^{−1}β, & if S is the Klein bottle Kb. \end{matrix}

Let M be the total space of an F -bundle over S defined by the projection p: M → S. Since S is aspherical, the long exact homotopy sequence for a fibration provides the short exact sequence:

1 ⟶ Π_{1}(F) ⟶^{i_{∗}} Π_{1}(M) ⟶^{p_{∗}} Π_{1}(S) ⟶ 1

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 57M50.

Key words and phrases. 4-manifold, fibre-bundle, geometry, group extension.

where i_{∗} is induced by the inclusion of the fibre, and p_{∗} is induced by the bundle projection. Viewing this sequence as a group extension the action of Π_{1}(S) on Π_{1}(F) determines a group homomorphism ϑ: Π_{1}(S) → Out(Π_{1}(F)). Below we obtain a presentation for Π_{1}(M) in terms of the images γ, δ of the generators of Π_{1}(F) under i_{∗}, and a fixed chosen pair α, β that project onto the generators of Π_{1}(S). Abusing notation the elements of this set {α, β, γ, δ} are refered to as the canonical generators of Π_{1}(M).

Definition. Two fibre-bundles p: M → S and p′: M′ → S′ are bundle-diffeomorphic when there is a pair of diffeomorphisms ϕ_{1}: M → M′ and ϕ_{2}: S → S′ such that ϕ_{2}p = p′ϕ_{1}. A bundle-diffeomorphism is a bundle-diffeomorphism over S when S = S′ and the mapping ϕ_{2} = id_{S}.

Definition. Two group extensions N↪^{ι}E↠^{ρ}G and N′↪^{ι′}E′↠^{ρ′}G′ are isomorphic when there is a pair of group isomorphisms ϕ_{1}: E → E′ and ϕ_{2}: G → G′ such that ϕ_{2}ρ = ρ′ϕ_{1}. An extension isomorphism is an extension-equivalence when the pair (N, G) = (N′, G′), the mapping ϕ_{2} = id_{G} and the restriction of ϕ_{1} to the image of i is the identity.

The Classification of Π_{1}(M)

The projection p is locally trivial; hence we may choose a small closed disk D ⊂ S so that the restriction of p to p^{−1}(Int(D)) is just the product projection. Now denoting M \ p^{−1}(Int(D)) by M_{0} we may write M as the adjunction space M_{0} ∪_{h} (F × D) where h: ∂ (F × D) → ∂(M_{0}) is a gluing homeomorphism of S^{1} ×F, whose restriction to S^{1} is the identity. In this way h defines a continuous loop of homeomorphisms of F and hence may be viewed as an element of Π_{1}(Homeo(F)).

We now examine the F -bundle that is p restricted to M_{0}. The base space S \ Int(D) is homotopy equivalent to S^{1} ∨ S^{1} and hence from the long homotopy sequence for a fibration we have:

1 ⟶ Π_{1}(F) ⟶ Π_{1}(M_{0}) ⟶ Π_{1}(S^{1} ∨ S^{1}) ⟶ 1.

Here Π_{1}(S^{1} ∨ S^{1}) is the free product ⟨α⟩ ∗ ⟨β⟩ so that this sequence splits and the group Π_{1}(M_{0}) is isomorphic to the semi-direct product Π_{1}(F) ⋊_{θ} (⟨α⟩ ∗ ⟨β⟩). The action θ: ⟨α⟩ ∗ ⟨β⟩ → Out(Π_{1}(F)) is the group homomorphism defined by θ = ϑϱ_{∗}, where ϱ_{∗} is induced by the inclusion S^{1} ∨ S^{1}→^{ϱ}S. Note in particular that the word ≀α, β≀_{S} ∈ (⟨α⟩ ∗ ⟨β⟩) is in the kernel of θ.

We now compute Π_{1}(M) by applying Van Kampen’s theorem to M = M_{0} ∪_{F ×S^{1}} (F × D). Writing η = ≀α, β≀_{S} we obtain a commutative diagram:

[figure: Π_{1}(∂(M_{0})); Π_{1}(F) ⋊_{θ} (⟨α⟩ ∗ ⟨β⟩); id⋊ϱ_{∗}; Π_{1}(F) × ⟨η⟩; ι; h_{∗}; ≀; Π_{1}(M); Π_{1}(F) × ⟨1⟩; id×0; i_{∗}]

Here the monomorphism i_{∗} is the familiar map induced by the inclusion of the fibre; and the mapping ι is the composite of h_{∗} and the canonical inclusion. Thus writing j_{∗} = (id ⋊ ϱ_{∗}) ∘ ι we have that

Π_{1}(M) ≅ (Π_{1}(F) ⋊_{θ} (⟨α⟩ ∗ ⟨β⟩)) /⟨j_{∗}((1, η))⟩.

Here the composite map j_{∗} is determined by h_{∗}, the map induced by h. Moreover the map ι factors through the cartesian product:

\xymatrix{Π_{1}(F) × ⟨η⟩ \ar@{^{(}->}[dr]^{ι} \ar@{-->}[r]^{f} & Π_{1}(F) × ⟨α⟩ ∗ ⟨β⟩ \ar@{->>}[d] \\  & Π_{1}(F) ⋊_{θ} (⟨α⟩ ∗ ⟨β⟩)}

where the mapping f precomposed with the projection on to the second factor, has image an infinite cyclic group generated by ≀α, β≀_{S} ∈ (⟨α⟩ ∗ ⟨β⟩). Furthermore the action of ≀α, β≀_{S} on Π_{1}(F) is trivial; and thus j_{∗}((1, η)) = γ^{a}δ^{b} ≀ α, β≀_{S}. Therefore writing δ^{−b}γ^{−a} = γ^{m}δ^{n} we obtain the presentation:

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}≀γ, δ≀_{F} = 1, & α(γ, δ)α^{−1} = ϑ(α)(γ, δ), \\ ≀α, β≀_{S} = γ^{m}δ^{n}, & β(γ, δ)β^{−1} = ϑ(β)(γ, δ). \end{matrix} ⟩ .

The pair ( \begin{matrix}m \\ n \end{matrix} ) ∈ ℤ^{2} represent the obstruction to constructing a cross-section to p and is refered to as the Euler number of the above presentation. Henceforth this group presentation shall be abreviated by the triple (ϑ(α), ϑ(β); ( \begin{matrix}m \\ n \end{matrix} )) known as the characteristic triple representing Π_{1}(M) with respect to the given presentation.

The question of which particular Euler numbers correspond to the fundamental group of such a bundle-space with given action ϑ is answered by considering the first homeotopy group of the fibre: Π_{1}(Homeo(F)).

In the case where F = T^{2}, we have that Π_{1}(Homeo(T^{2})) ≅ ℤ^{2} [Ha1]; in such a way that every obstruction ( \begin{matrix}m \\ n \end{matrix} ) ∈ ℤ^{2} is realized by some gluing homeomorphism h as above [FS]. Moreover every action ϑ can be realized as the monodromy of such a bundle-projection. Thus as the total-space M is aspherical, the set Hom (Π_{1}(S), Out(Π_{1}(F)) × ℤ^{2} modulo transformations induced by changes in the base and and fibre groups characterizes the homotopy type of all T^{2}-bundle-spaces over S. However when the fibre is F = Kb, many such ( \begin{matrix}m \\ n \end{matrix} ) ∈ ℤ^{2} do not define torsion-free groups. In particular Π_{1}(Homeo(Kb)) ≅ ℤ [Ha2]. Thus as above the homotopy type of Kb-bundles over S is characterized by the set Hom (Π_{1}(S), Out(Π_{1}(F)) × ℤ modulo transformations induced by automorphisms of the base and fibre groups. In light of the consequences of Dehn-Nielsen Realization Theorem [ZVC] we shall be able to determine how ℤ embeds in ℤ^{2} and hence determine the possible Euler numbers associated with each given action ϑ that give rise to bona fide bundle-space groups.

Although we are interested in the bundle-diffeomorphism type of M it is interesting to know to what extent Π_{1}(M) determines the diffeomorphism type of M.

The Diffeomorphism Type of M

By exploiting the S^{1}-bundle over S^{1} structure of the base-space S, thereby viewing such 4-manifolds as bundle-spaces over S^{1} with fibre some small Seifert manifold, Ohba [Oh] has shown for the following pairs (F, S): (T^{2}, T^{2}), (Kb, T^{2}) and (Kb, Kb) the homotopy type classes of M are in bijection with the diffeomorphism type classes of M. (The result for the case (F, S) = (T^{2}, T^{2}) was originally proved by [F,S] through different means.) Thus in these cases two non-bundle-diffeomorphic total-spaces cannot have the same characteristic triple.

In the remaining case where the pair (F, S) = (T^{2}, Kb), such a bundle ξ has a two-fold covering ð^{∗}(ξ) that is a T^{2}-bundle over T^{2} induced by the orientation double covering ð: T^{2} ↠ Kb. Therefore from above, the fundamental group of the total space M̃ of the bundle ð^{∗}(ξ) determines the diffeomorphism type of M̃. Of course any one T^{2}-bundle over T^{2} may cover many non-bundle-diffeomorphic (or even non-diffeomorphic) T^{2}-bundle-spaces over Kb. In any case the techniques to follow will guarantee that the characteristic triple provides a classification of bundle-diffeomorphism type, as this invariant not only describes the isomorphism class of Π_{1}(M) but also the induced epimorphism p_{∗}: Π_{1}(M) → Π_{1}(S).

The Bundle-Diffeomorphism Type of M

A bundle-diffeomorphism (ϕ_{1}, ϕ_{2}) induces a triple of group automorphisms defining an isomorphism of the corresponding homotopy sequences:

\xymatrix{Π_{1}(F) \ar@{^{(}->}[r] \ar[d]^{≀}_{ϕ_{0}} & Π_{1}(M) \ar@{->>}[r]^{p_{∗}} \ar[d]^{≀}_{(ϕ_{1})_{∗}} & Π_{1}(S) \ar[d]^{≀}_{(ϕ_{2})_{∗}} \\ Π_{1}(F′) \ar@{^{(}->}[r] & Π_{1}(M′) \ar@{->>}[r]^{p′_{∗}} & Π_{1}(S′).}

In fact even more is true, as is outlined in the following three propositions:

Proposition A. Let ξ be an F -bundle with projection p: M → S. Then for any isomorphism ψ_{S}: Π_{1}(S′) → Π_{1}(S) there is a bundle-diffeomorphism:

\xymatrix{M′ \ar@{->>}[r]^{p′} \ar[d]_{ψ̂_{M}}^{≀} & S′ \ar[d]_{ψ̂_{S}}^{≀} \\ M \ar@{->>}[r]^{p} & S}

for some F -bundle p′: M′ → S′; where ψ̂_{S} is a diffeomorphism realizing ψ_{S}. If (ϑ(α), ϑ(β); ( \begin{matrix}m \\ n \end{matrix} )) is the characteristic triple of p then (ϑ(ψ_{S}(α)), ϑ(ψ_{S}(β)); ({m′ \atop n′})) is the characteristic triple of p′, where the vector ({m′ \atop n′}) corresponds to the word ψ_{S}(≀α, β≀_{S}) = γ^{m′}δ^{n′}.

proof. By the Dehn-Nielsen Theorem there is a diffeomorphism ψ̂_{S}: S′ → S realizing ψ_{S}. Define p′: M′ → S′ to be the bundle (ψ̂_{S})^{∗}(ξ), induced by ψ̂_{S}.

□ Proposition B. Let ξ be an F -bundle with projection p: M → S. Then for any isomorphism ψ_{F}: Π_{1}(F) → Π_{1}(F′) there is an bundle-diffeomorphism over S:

\xymatrix{M \ar@{->>}[dr]^{p} \ar[dd]_{ψ̂_{M}}^{≀} \\  & S \\ M′ \ar@{->>}[ur]^{p′}}

for some F -bundle p′: M′ → S; where the restriction of ψ̂_{M} to the fibre over the basepoint of S is a diffeomorphism realizing ψ_{F}. If (ϑ(α), ϑ(β); ( \begin{matrix}m \\ n \end{matrix} )) is the characteristic triple of p then (κϑ(α)κ^{−1}, κϑ(β)κ^{−1}; ({m′ \atop n′})) is the characteristic triple of p′, where κ represents the restriction of (ψ̂_{M})_{∗} to ker(p_{∗}) in Out(Π_{1}(F)); and ({m′ \atop n′}) is the vector defined by the word κ(γ^{m}δ^{n}) = γ^{m′}δ^{n′}.

proof. Let ψ̂_{F}: F → F′ be a diffeomorphism realizing ψ_{F} and consider the mapping

ϕ: F × 𝔉_{S} ⟶^{ϕ̂_{F} ×id}F′ × 𝔉_{S}. This induces a bundle-diffeomorphism ϕ̃: M(0) → M(0)′ where the bundle-space M(0)′ is represented by the triple (κϑ(α)κ^{−1}, κϑ(β)κ^{−1}; ({0 \atop 0})), with κ ∈ Out(Π_{1}(F)) as above. Then writing M = M(0)\p_{0}^{−1}(Int(D))∪_{h} D×F we may construct M’ as the adjunction space M(0)′ \(p′_{0})^{−1}(Int(D))∪_{h′} D ×F′; where the gluing map h′: (p′_{0})^{−1}(∂D) → D × F is defined by h′ = ϕ̃ ∘ h ∘ ϕ̃^{−1}. Whence a bundle-diffeomorphism is given by:

ψ̂_{M} = \{ \begin{matrix}ϕ̃, & on M(0) \ p_{0}^{−1}(Int(D)), \\ id × ψ̂_{F} , & on D × F . \end{matrix}

□

Proposition C. Let ξ be an F -bundle over S. Then for any extension equivalence ψ: Π_{1}(M) → Π_{1}(M′) of the associated exact sequence for ξ, there is a bundlediffeomorphism ψ̂: M→^{∼}M′ over S realizing ψ, for some F -bundle p: M′ → S. If (ϑ(α), ϑ(β); ( \begin{matrix}m \\ n \end{matrix} )) is the characteristic triple of p then (ϑ(α), ϑ(β); ({m′ \atop n′})) is the characteristic triple of p′ where the vector ({m′ \atop n′}) is defined by the word ≀υ(α), υ(β)≀_{S} = γ^{m′}δ^{n′} and υ is the set-theoretic cross-section to p determined by ψ and the canonical generators of Π_{1}(M).

proof. Let D ⊂ S be a small disk that lifts to the product D×F under p. Then S\D retracts to S^{1}∨S^{1} and therefore the restricted bundle projection M \p^{−1}(Int(D)) → S \ D admits a cross-section. The base-point ∗ of the fibre F determines a such a section σ: S \ D ↪ M \ p^{−1}(Int(D)).

Extention equivalences correspond to a change ς ↦ ς′ of set-theoretic crosssection ς̂: Π_{1}(S) → Π_{1}(M) to p_{∗}; which in turn correspond to mappings υ of the free group on the generators of Π_{1}(S) into Π_{1}(M \ p^{−1}(Int(D))). Any such a mapping υ is realized by a cross-section of the restricted bundle-projection as given above. By the homogeniety of F we may change its base-point via some diffeomorphism (F, ∗)→^{ς}(F, ∗′) that is isotopic to the identity. As for proposition B this self-diffeomorphism defines a diffeomorphism ϕ on F × 𝔉_{S} which induces a diffeomorphism ϕ̂ between M and M(0)′ \ (p′_{0})^{−1}(Int(D)) ∪_{h′} D × F′ where h′ = ϕ̃hϕ̃^{−1}, giving the desired bundle-diffeomorphism. □

In view of the above three propositions and the preceding comment we are lead to the crucial theorem inspiring our proposed classification.

Theorem. Two F -bundles over S are bundle-diffeomorphic precisely when their associated exact sequences are isomorphic.

Remark. This theorem was essentially proved in [Hi] chapter IV where it is shown that two surface-bundles over a surface with aspherical base and fibre are homotopy equivalent precisely when the associated group extensions of the bundles are isomorphic.

Proof. Let ξ′ and ξ be two F -bundles over S with a given extension isomorphism (ϕ_{0}, ϕ_{1}, ϕ_{2}) between their associated homotopy sequences. By proposition A with ϕ_{2} ∈ Aut(Π_{1}(S)) there exists a bundle-diffeomorphism ξ′′→^{∼}ξ realizing the corresponding extension isomorphism (id, ϕ̃_{2}, ϕ_{2}) where ϕ̃_{2} is the pullback along p_{∗} of ϕ_{2}.

Now applying proposition B to the bundle ξ′′ with ϕ_{0} ∈ Aut(Π_{1}(F)) we obtain a bundle ξ′′′ that is bundle-diffeomorphic to ξ′′ with associated extension isomorphism (ϕ_{0}, ϕ̃_{0}, id). In which case the associated sequence of ξ′′′ is isomorphic to the associated sequence of ξ via the isomorphism (ϕ_{0}, ϕ̃_{0}ϕ̃_{2}, ϕ_{2}). Therefore we have an extension equivalence between the associated exact sequences of ξ′ and ξ′′′ given by the isomorphism ϕ_{1}^{−1}ϕ̃_{0}ϕ̃_{2}: Π_{1}(M′′′)→^{∼}Π_{1}(M′).

Hence by proposition C the bundles ξ′ and ξ′′′ are bundle-diffeomorphic. Thus by transitivity of the bundle-diffeomorphism relation it follows that ξ is bundlediffeomorphic to ξ′. □

As an aid in attaining bundle-diffeomorphism classes of such fibre bundles we will make use of the following well known theorem about group extensions:

Theorem. Let Ξ be the group extension:

1 ⟶ N ⟶ E ⟶ G ⟶ 1

with action ϑ: → Out(N). Then equivalence classes of Ξ are in bijection with the elements of H_{κ}^{2}(G; Z(N)), where the center of N, Z(N) is regarded as a G-module via the action κ: G → Aut(Z(N)) induced by ϑ.

In particular it is well known that extensions with distinct actions vartheta are non-equivalent. Furthermore the question of existence of an extension Ξ with given action ϑ is decided by examination of a correspondence between the homomorphism ϑ and an element of H_{κ}^{3}(G; Z(N)). Such an extension Ξ is realised with respect to the action ϑ precisely when ϑ corresponds to the zero element of H_{κ}^{3}(G; Z(N)). In our case the groups in question G all have cohomological dimension 2 and thus all possible actions ϑ are realized. Thus bundle-diffeomorphism classes a priori correspond to some quotient of H_{κ}^{2}(Π_{1}(S); Z(Π_{1}(F))); namely by those transformations on the base and fibre, as extension equivalences correspond to a change of set-theoretic cross-section to the group surjection p_{∗}. We refer the reader unfamiliar with group cohomology to [Ro] and [Br].

We calculate the cardinality of extension equivalence classes as follows. By duality we have that

H_{κ}^{2}(Π_{1}(S); Z(Π_{1}(F))) ≅ H_{0}^{w_{1}(S)⊗κ}(Π_{1}(S); ℤ ⊗_{ℤ} Z(Π_{1}(F))).

Where the action of Π_{1}(S) on the orientation module ℤ is via the orientation character w_{1}(S): Π_{1}(S) → {±1}. Thus writing Π_{1}(Kb) = ⟨γ, δ⟩ we have that Z(Π_{1}(Kb)) = ⟨γ^{2}⟩ ≅ ℤ so that:

H_{κ}^{2}(Π_{1}(S); Z(Π_{1}(F))) ≅ \{ \begin{matrix}ℤ^{2}/⟨κ(g) − w_{1}(S)(g)⟩, & if F = T^{2}; or \\ ℤ/⟨κ(g) − w_{1}(S)(g)⟩, & if F = Kb. \end{matrix}

where the elements g range over Π_{1}(S).

As we now possess the machinery to classify such bundle-spaces up to bundlediffeomorphism, we proceed as proclaimed.

The Classification of Non-Orientable Total Spaces of T^{2}-Bundles over T^{2} and their Geometries

The classification of T^{2}-bundles with orientable total spaces together with a given geometric structure was achieved by Ue [Ue]. Below in the spirit of [Ue] we complete the classification of T^{2}−bundles over T^{2} to intially obtain a list of such bundle-spaces that are non-orientable. We then proceed to show that all of these 4manifolds are geometric by explicitly giving faithful, discrete representations from the fundamental group into the appropriate isometry group. By writing M = (A, B; ( \begin{matrix}a \\ b \end{matrix} )) we are refering to the bundle-space with fundamental group

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h)A, \\ [α, β] = l^{a}h^{b}, & β(l, h)β^{−1} = (l, h)B. \end{matrix} ⟩;

where A and B are matrices in Aut(Π_{1}(T^{2})) = GL(2, ℤ). The results are captured in the following:

Theorem. Let M be the non-orientable total space of a T^{2}-bundle over T^{2} with first Betti number β_{1}(M). Then M is bundle diffeomorphic to one of the geometric 4-manifolds listed below:

Total Space  Geometric Type

β_{1}(M) = 3  \begin{matrix}(({−1 \atop 0} {0 \atop 1}), I; ({(1) \atop 0})) & E^{4} \\ (({−1 \atop 0} {1 \atop 1}), I; ({0 \atop 0})) & E^{4} \end{matrix}

β_{1}(M) = 2

(( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}0 \\ (1) \end{matrix} ))  E^{4}
(( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}1 \\ 1 \end{matrix} ))  E^{4}
(( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}0 \\ (1) \end{matrix} ))  E^{4}
(( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}(1) \\ n \end{matrix}))_{n∈ℕ\\{0\}}  Nil^{3} × E
(( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}0 \\ n \end{matrix}))_{n∈ℕ\\{0\}}  Nil^{3} × E
(A, ±I; ( \begin{matrix}a \\ b \end{matrix}), det(A) = −1, rank(A − I) = 2)  Sol^{3} × E

Table 1.

Remark. Each characteristic triple listed gives a distinct bundle-space with the exception of the infinite family of Sol^{3} × E-manifolds. In this case one may obtain further refinement of bundle-space representation by observing that there is a bijection between similarity classes of such a matrix A and the class number of the ring ℤ[ζ]; where ζ is an eigenvalue of A. In particular there are finitely similarity classes corresponding to each pair of eigenvalues ζ, ζ^{−1} of A. For discussion of such matters we refer the reader to [Ne].

For future reference we include a listing [Ue1] of all such orientable bundle-spaces.

Theorem. Let M be the orientable total space of a T^{2}-bundle over T^{2} with first Betti number β_{1}(M). Then M is bundle diffeomorphic to one of the geometric 4manifolds listed below:

Total Space  Geometric Type

β_{1}(M) = 4  (I, I; ({0 \atop 0}))  E^{4}

β_{1}(M) = 3  (I, I; ({1 \atop 0}))  E^{4}

β_{1}(M) = 2

(( \begin{matrix}−1 & −1 \\ 1 & 0 \end{matrix} ) , I; ( \begin{matrix}0 \\ 0 \end{matrix} ))  E^{4}
(( \begin{matrix}−1 & −1 \\ 1 & 0 \end{matrix} ) , −I; ( \begin{matrix}0 \\ 1 \end{matrix} ))  E^{4}
(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , I; ( \begin{matrix}0 \\ (1) \end{matrix} ))  E^{4}
(−I, I; ( \begin{matrix}(1) \\ 0 \end{matrix} ))  E^{4}
(( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}a \\ b \end{matrix}))_{λ,b≠0}  Nil^{4}
(( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , −I; ({(1) \atop (1)}))_{λ≠0}  Nil^{3} × E
(( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ) , I; ({(1) \atop (1)}))_{λ≠0}  Nil^{3} × E
(A, ±I; ( \begin{matrix}a \\ b \end{matrix}), det(A) = 1, |trace(A)| > 2)  Sol^{3} × E

Table 2.

Note. Some of the representations as they appear in Table 2. have been slightly altered from their original form in [Ue1] so as to conform to our conventions.

The following three classes of fibre preserving diffeomorphims correspond to the respective transformations of the algebraic representation (A, B; \binom{a}{b}) of the bundle space M:

(1) Change of framing of the fibre

(l, h) ↦ (l, h)P, ∃P ∈ GL(2, ℤ).
(A, B; ( \begin{matrix}a \\ b \end{matrix} )) ↦ (P^{−1}AP, P^{−1}BP; P^{−1}( \begin{matrix}a \\ b \end{matrix} )).

(2) Change of base of the base space

(α, β) ↦ (α, β)P, ∃P = ( \begin{matrix}p & q \\ r & s \end{matrix} ) ∈ GL(2, ℤ).
(A, B; ( \begin{matrix}a \\ b \end{matrix} )) ↦ (A^{p}B^{r}, A^{q}B^{s}; det(P)( \begin{matrix}a \\ b \end{matrix} )).

(3) Change of cross-section

(α, β) ↦ (l^{s}h^{t}α, l^{p}h^{q}β), where t, s, p, q ∈ ℤ are free.
(A, B; ( \begin{matrix}a \\ b \end{matrix} )) ↦ (A, B; ( \begin{matrix}a \\ b \end{matrix} ) + (A^{−1} − I)( \begin{matrix}p \\ q \end{matrix}) + (I − B^{−1}) ( \begin{matrix}s \\ t \end{matrix} )).

A constructive realization of the above bundle isomorphisms from the algebraic transformations may be found in [FS].

In this case equivalence classes of triples modulo transformations of type (3) are in bijection with elements of H^{2}((Z)^{2}; ℤ^{2}) ≅ ℤ^{⊭}/⟨A − I, B − I⟩.

The bundle-space M is orientable precisely when det(A) = 1 = det(B). Below we shall only consider the case det(A) = −1 = − det(B). For if det(A) = 1 and det(B) = −1 we may interchange A and B while if det(A) = −1 = det(B) we may replace B by AB. On examination of the presentation for H_{1}(M) we see that β_{1}(M) = 4 − rank(A − I, B − I, \binom{a}{b}). Furthermore β_{1}(M) = 4 if and only if M ≅ T^{4} = T^{2} × T^{2}. We proceed by exhausting all possible cases for the conjugacy classes of representation (A, B; \binom{a}{b}) with respect to possible values of β_{1}(M). In doing this we will make frequent implicit use of the following lemma. The symbol (1) is used to denote a matrix entry that is either 1 or 0.

Lemma. The conjugacy class of a matrix A in GL(2, ℤ) is determined as follows: (1) If rank(A − I) = 0 then A = I.

(2) If rank(A − I) = 1 then A is conjugate to either ( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} )_{λ≠0} if det(A) = 1 or

precisely one of the two matrices ( \begin{matrix}−1 & (1) \\ 0 & 1 \end{matrix} ) if det(A) = −1.

(3) Let rank(A − I) = 2 and det(A) = 1.

a) If |trace(A)| < 2 then A is conjugate to precisely one of the three

matrices ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), ( \begin{matrix}−1 & −1 \\ 1 & 0 \end{matrix} ) or ( \begin{matrix}1 & −1 \\ 1 & 0 \end{matrix} ).

b) If |trace(A)| = 2 then A = ( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ).

c) If |trace(A)| > 2 then A has eigenvalues ζ, 1/ζ for some ζ ∈ ℝ \ ℚ.

(4) If rank(A − I) = 2 and det(A) = −1 then A has eigenvalues ζ, −1/ζ for some ζ ∈ ℝ \ ℚ.

proof: The first claim is clear. A proof of (2) may be found in [FS]. The proof of (3)a) and b) comes from examining the possible characteristic polynomials λ^{2} − trace(A)λ + det(A) of such matrices A and recognising that these polynomials are in fact class functions. The proofs of (3)c) and (4) follow quickly from inspection of the possible characteristic polynomials of such matrices. □

As a consequence of the following theorem due to Wall [Wa] our task: to prove that each characteristic triple appearing in table 1 represents a distinct bundlediffeomorphism class; is reduced to the problem of distinguishing between only those representations admitting common geometries.

Theorem. Two closed 4-manifolds admitting distinct geometries are not homotopy equivalent.

Henceforth in initialy proving that Table 1. is a complete listing we assume the given characteristic triples admit the geometries as claimed.

Proof of the theorem: From our convention that det(A) = −1 = − det(B) we must have that A ≠ I so that rank(A − I) is either 1 or 2.

A) The case β_{1}(M) = 3 ⟹ rank(A − I, B − I; \binom{a}{b}) = 1.

Thus rank(A − I) = 1 and so A ∼ ({−1 \atop 0} {(1) \atop 1}) and we must have b = 0. In this case AB = BA and det(B) = 1 together imply B = ±I. But rank(B − I) cannot exceed 1 so we must have B = I. Under an appropriate change of cross-section we may transform \binom{a}{0} to \binom{0}{0} or \binom{(1)}{0} depending upon whether the value of the entry (1) in A is respectively either 1 or 0. That is we obtain the three representations:

(( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}(1) \\ 0 \end{matrix} )), (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}0 \\ 0 \end{matrix} )).

However there is a group isomorphism between (\binom{−1 1}{0 1}, I; \binom{0}{0}) and (\binom{−1 0}{0 1}, I; \binom{1}{0}) fixing the generators α and l and mapping β to h^{−1}. Hence following [Oh], these two representations represent two fibrations with diffeomorphic total spaces. But since ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) is not conjugate to ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) any such diffeomorphism cannot be realized by a bundle diffeomorphism. Thus, as bundle spaces these representations are distinct. The remaining two groups are distinguished by their abelianizations and hence represent distinct manifolds.

B) The case β_{1}(M) = 2 ⟹ rank(A − I, B − I; \binom{a}{b}) = 2.

B_{1}) rank(A − I) = 1 ⟹ A ∼ ({−1 \atop 0} {(1) \atop 1}) and as before we must have B = ±I. If B = I then rank(B − I) = 0 and we must have b ≠ 0. As before we may transform \binom{a}{b} by an appropriate change of cross-section to either \binom{0}{b} or \binom{(1)}{b} depending upon whether the value of the entry (1) in A is respectively either 1 or 0. Furthermore conjugating A and B in the first instance by −I and in the second instance by ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) we may assume that b is positive. That is we obtain the three infinite families of representations:

(( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}(1) \\ n \end{matrix} ))_{n∈ℕ\\{0\}}, (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}0 \\ n \end{matrix} ))_{n∈ℕ\\{0\}}.

After fixing the two pairs (A, B) as above, the above families of triples correspond to group extension equivalence classes since

H^{2}(ℤ^{2}; ℤ^{2}) ≅ \{ \begin{matrix}ℤ ⊕ ℤ/2ℤ, & for (A, B = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , I).; \\ ℤ, & for (A, B = ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , I). \end{matrix}

Abelianization of these representations shows they are mutualy non-isomorphic and hence represent distinct manifolds. If B = −I then under an appropriate change of cross-section we may assume \binom{a}{b} = \binom{(1)}{(1)} or \binom{0}{(1)} depending on whether the respective value of the entry (1) in A is 0 or 1. That is we obtain the following six representations:

(( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}(1) \\ (1) \end{matrix} )), (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}0 \\ (1) \end{matrix} )).

Note that

H^{2}(ℤ^{2}; ℤ^{2}) ≅ \{ \begin{matrix}ℤ/2ℤ ⊕ ℤ/2ℤ, & for (A, B = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , −I); \\ ℤ/2ℤ, & for (A, B = ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , −I); \end{matrix}

so that these triples are seen to capture all equivalence classes of the possible group extensions with the action (A, B) fixed as above. However we may transform (\binom{−1 0}{0 1}, −I; \binom{0}{1}) to (\binom{−1 0}{0 1}, −I; \binom{1}{0}) by replacing A by AB and then conjugating by \binom{0 1}{1 0}; yielding an extra extension isomorphism that is not an equivalence. The remaining five representations are distinct and may be distinguished as follows: Let

M_{1} = (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ), −I; ( \begin{matrix}0 \\ 0 \end{matrix} )), M_{2} = (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ), −I; ( \begin{matrix}0 \\ 1 \end{matrix} )), M_{3} = (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ), −I; ( \begin{matrix}1 \\ 1 \end{matrix} )),
M_{4} = (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ), −I; ( \begin{matrix}0 \\ 0 \end{matrix} )), M_{5} = (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ), −I; ( \begin{matrix}0 \\ 1 \end{matrix} )).

Then abelianizing we have:

H_{1}(M_{1}) = ℤ^{2} ⊕ (ℤ/2ℤ)^{2} ,
H_{1}(M_{2}) = ℤ^{2} ⊕ ℤ/2ℤ = H_{1}(M_{3}) = H_{1}(M_{4}),
H_{1}(M_{5}) = ℤ^{2};

distinguishing M_{1} and M_{5}. Furthermore the commutator subgroups Π_{1}(M_{2})′ ≅ ℤ^{2} ≅ Π_{1}(M_{4})′ and Π_{1}(M_{3})′ ≅ ℤ are characteristic and hence distinguish M_{3}.

Finally the bundle-spaces M_{2} and M_{4} must be distinct as the centers of their fundamental groups are non-isomorphic. In particular Z(Π_{1}(M_{2})) = ⟨β^{2}⟩ ≅ ℤ, whilst Z(Π_{1}(M)_{4}) = ⟨α^{2}, β^{2}⟩ ≅ ℤ^{2}.

B_{12}) rank(B − I) = 1 ⟹ B ∼ ({1 \atop 0} {λ \atop 1})_{λ≠0} . But as before, the relation [A, B] = I

gives λ = 0 and this case does not represent a bundle-space.

B_{2}) rank(A − I) = 2 ⟹ A has eigenvalues ζ, −1/ζ ∈ ℝ \ ℚ.

B_{21}) rank(B − I) = 0 ⟹ B = I and we obtain the infinite family of manifolds:

(A, I; ( \begin{matrix}a \\ b \end{matrix} ), det(A) = −1, rank(A − I) = 2)

B_{22}) rank(B − I) = 1 ⟹ B ∼ ({1 \atop 0} {λ \atop 1})_{λ≠0}. Writing B = I + N the relation

AB = BA implies that A commutes with N, contradicting det(A) = −1. Thus this representation does not represent a bundle-space.

B_{23}) rank(B − I) = 2. If |trace(B)| < 2 then B is conjugate to one of ({0 \atop 1} {−1 \atop 0}), ({−1 \atop 1} {−1 \atop 0}) or ({1 \atop 1} {−1 \atop 0}). An easy calculation together with the inequality x^{2} +y^{2} ≥ |xy|, ∀x, y ∈ ℤ shows that the centralizer of any one of the above matricies in GL(2, ℤ) only contains matrices of positive determinant; contradicting det(A) = −1. If |trace(B)| = 2 then B ∼ \binom{−1 λ}{0 −1}. Writing B = −I + N the relation AB = BA yields a contradiction as in the case B_{22}) unless λ = 0 so that we obtain the infinite family of manifolds:

(A, −I; ( \begin{matrix}a \\ b \end{matrix} ), det(A) = −1, rank(A − I) = 2)

Remark. Since H^{2}(ℤ^{2}; ℤ^{2}) is some quotient of ℤ/2ℤ ⊕ ℤ/2ℤ there are at most four such bundle-spaces correspondind to each choice of A.

If |trace(B)| > 2 then by Lemma 3. [FS] the pair (A, B) is equivalent to a pair (C, ±I) where the eigenvalues of C are some q^{th} roots of the respective eigenvalues of A and hence are irrational. That is we obtain the infinite family of manifolds already covered in case B_{21} and this case above.

And we have shown that Table 1. is a complete list of all non-orientable 4manifolds that arise as total spaces of T^{2}-bundles over T^{2} up to bundle diffeomorphism. Below we show that they admit the geometries as claimed above.

The isometry group I som(E^{4}) of flat 4-space is ℝ^{4} ⋊ O(4) where the action of O(4) on ℝ^{4} is the standard left matrix multiplication.

Lemma. A representation Φ: Π_{1}(M) → I som(E^{4}) is faithful and discrete with cocompact image precisely when the restriction of Φ to some normal subgroup A ◁ Π_{1}(M) of finite index is faithful and discrete with cocompact image.

Remark. If indeed M admits an E^{4}-geometry we are guaranteed of the existence of such a subgroup A. For if Π_{1}(M) embeds discretely into ℝ^{4} ⋊ O(4) via Φ with compact quotient, then the image A of Φ(Π_{1}(M)) in the translation group ℝ^{4} must be discrete and cocompact; and thus free abelian of rank 4. Moreover the quotient ℝ^{4}/A ≃ T^{4} is a compact manifold that finitely covers M; and therefore the index of A in Π_{1}(M) is finite. In fact flat n-manifold groups (also known as Bieberbach groups) are characterized by being torsion free and virtually ℤ^{n}. See for example [Ch]. proof: It is clear that if Π_{1}(M) embeds faithfully and discretely into I som(E^{4}) then so does A. To see that cocompactness of Φ(Π_{1}(M)) implies that Φ(A) is cocompact note we have an n−fold covering space projection [figure: E^{4}/Φ(A); p; E^{4}/Φ(Π_{1}(M));] where n is the index of A in Π_{1}(M). Take an open covering of the total space and view each non-regular set in this cover as a union of regular sets. Then since p is a local homeomorphism and hence an open map, any open covering of the total space projects onto an open covering of the base space. Whence we may pass to a regular finite open covering which may be lifted to obtain a finite open regular covering of E^{4}/Φ(A). Now since p is a local homeomorphism we may form appropriate unions of these regular sets to obtain a finite subcovering of the original covering.

Below we prove the reverse implication. Since M is the total space of a surfacebundle over a surface S ≠ ℝℙ^{2} with fibre F ≠ ℝℙ^{2}, Π_{1}(M) is torsion free. To see this suppose g ∈ Π_{1}(M) has finite order. Then examining the exact homotopy sequence for a fibration:

\xymatrix{1 \ar[r] & Π_{1}(F) \ar[r]^{i_{∗}} & Π_{1}(M) \ar[r]^{p_{∗}} & Π_{1}(S) \ar[r] & 1.}

we must have that g ∈ ker(p_{∗}) since Π_{1}(S) is torsion free. Therefore g ∈ im(i_{∗}) and hence g = 1 as Π_{1}(F) is torsion free.

We show that Φ is faithful. Let K = Ker(Φ) so that we have a commutative diagram as shown below:

[figure: A ∩ K; K; K/(A ∩ K); ψ; A; Φ|A; Π_{1}(M); Φ; Π_{1}(M)/A; I som(E^{4})]

where ψ is the induced map.

If x ∈ A ∩ K is non-trivial then Φ(x) = 1 since x ∈ K; but Φ(x) ≠ 1 since x ∈ A so we must have A ∩ K = {1} and ψ is a monomorphism. However since Π_{1}(M) is torsion free, K must be torsion free and ψ embeds K in the finite group Π_{1}(M)/A. Thus K is trivial and the representation Φ is faithful. Furthermore it is clear that Φ(A) cocompact implies Φ(Π_{1}(M)) is cocompact. And finally if Φ(A) is discrete and Φ(Π_{1}(M)) is not discrete then there is a point x⃗ ∈ ℝ^{4} and an infinite squence (g_{i}) of elements of Φ(Π_{1}(M)) such that g_{i}(x⃗) converges to x⃗. Now since the index of Φ(A) in Φ(Π_{1}(M)) is finite at least one of the cosets of Φ(A) in Φ(Π_{1}(M)) must contain infintely many elements of this sequence. Hence without loss of generality we may assume g_{i} = a_{i}g where a_{i} ∈ Φ(A) and g ∈ Φ(Π_{1}(M)) is fixed. Thus if d(−, −) is the euclidean metric on ℝ^{4} then d(a_{n}g(x⃗), x⃗) < ϵ/2 for all fixed ϵ > 0 and all n > N_{0}(ϵ). Hence by the triangle inequality d(a_{n}g(x⃗), a_{m}g(x⃗)) ≤ ϵ for all n, m > N_{0}. But since Φ(A) acts isometrically we have that d(g(x⃗), a_{n}^{−1}a_{m}(g(x⃗))) ≤ ϵ, contradicting the assumption that Φ(A) is discrete. Thus Φ(Π_{1}(M)) is discrete and the proof is complete. □

The bundle-space (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h), \\ [α, β] = 1, & β(l, h)β^{−1} = (l, h). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, l, h⟩ of Π_{1}(M) has index 2 and maps faithfully into the

translation group ℝ^{4} under Φ with discrete image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

This image is indeed cocompact as E^{4}/Φ(A) ≃ ℝ^{4}/ℤ^{4} = T^{4} a compact manifold. Thus this representation is both faithful and discrete with cocompact image so that (({−1 \atop 0} {0 \atop 1}) , I; \binom{0}{0}) admits an E^{4} geometry.

The bundle-spaces (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}0 \\ 0 \end{matrix} )) and (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}1 \\ 0 \end{matrix} )) admit an E^{4} geometry.

We show the first representation is geometric.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, lh), \\ [α, β] = 1, & β(l, h)β^{−1} = (l, h). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

Now by a change of ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{3} +2e_{4}⟩ we have a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( 0 ; 0 ; − \frac{1}{2} ; \frac{1}{2} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

As before the subgroup A⟨α^{2}, β, l, h⟩ of Π_{1}(M) has index 2 and maps faithfully

into ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that so that (({−1 \atop 0} {1 \atop 1}) , I; \binom{0}{0}) admits a geometry of type E^{4}. Moreover since

(( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}1 \\ 0 \end{matrix} )) is abstractly group isomorphic to (({−1 \atop 0} {1 \atop 1}) , I; \binom{0}{0}), the compo-

sition of such an isomorphism with Φ above shows that (({−1 \atop 0} {1 \atop 1}) , I; \binom{1}{0}) admits a geometry of type E^{4}.

The bundle-space (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h), \\ [α, β] = 1, & β(l, h)β^{−1} = (l^{−1}, h^{−1}). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4}

under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus

this representation is both faithful and discrete with cocompact image so that (({−1 \atop 0} {0 \atop 1}) , −I; \binom{0}{0}) admits an E^{4} geometry.

The bundle-space (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}0 \\ 1 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h), \\ [α, β] = h, & β(l, h)β^{−1} = (l^{−1}, h^{−1}). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ −1 & 0 & 0 & −1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨2e_{1} −e_{4}, e_{2}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ α & ↦ (( \frac{1}{2} 0 0 \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

16 ROBIN J. COBB

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}1 \\ 0 \\ 0 \\ 1 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus this representation is both faithful and discrete with cocompact image so that (({−1 \atop 0} {0 \atop 1}) , −I; \binom{0}{1}) admits an E^{4} geometry. The bundle-space (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}1 \\ 1 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h), \\ [α, β] = lh, & β(l, h)β^{−1} = (l^{−1}, h^{−1}). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , /I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & −1 & 0 \\ −1 & 0 & 0 & −1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨2e_{1} − e_{4}, 2e_{2} + e_{4}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 ; \frac{1}{2} ; − \frac{1}{2} ; 0 ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ α & ↦ (( \frac{1}{2} 0 0 \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}1 \\ 0 \\ 0 \\ 1 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus this representation is both faithful and discrete with cocompact image so that (({−1 \atop 0} {0 \atop 1}) , −I; \binom{1}{1}) admits an E^{4} geometry. The bundle-space (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, lh), \\ [α, β] = 1, & β(l, h)β^{−1} = (l^{−1}, h^{−1}). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{3} +2e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( 0 ; 0 ; − \frac{1}{2} ; \frac{1}{2} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4}

under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus

this representation is both faithful and discrete with cocompact image so that (({−1 \atop 0} {1 \atop 1}) , −I; \binom{0}{0}) admits an E^{4} geometry.

The bundle-space (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}0 \\ 1 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, lh), \\ [α, β] = h, & β(l, h)β^{−1} = (l^{−1}, h^{−1}). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℝ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ − \frac{1}{2} & − \frac{1}{2} & −1 & 0 \\ −1 & 0 & 0 & −1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}−1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & − \frac{1}{2} & −1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨−2e_{1} + 2e_{2} + e_{4}, −4e_{2} + e_{3}, e_{3}, e_{3} + 2e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( 0 ; 0 ; − \frac{1}{2} ; \frac{1}{2} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 ; − \frac{1}{4} ; \frac{1}{4} ; 0 ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ α & ↦ (( \frac{1}{2} ; − \frac{1}{4} ; 0 ; \frac{1}{4} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

Then the subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and embeds discretely

into ℝ^{4} under Φ as ⟨( 1 ; − \frac{1}{2} ; 0 ; \frac{1}{2} ) , ( \begin{matrix}0 \\ − \frac{1}{2} \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( 0 ; 0 ; − \frac{1}{2} ; \frac{1}{2} )⟩. As before these four vectors

are linearly independent whence Φ(A) is cocompact. and (({−1 \atop 0} {1 \atop 1}) , −I; \binom{0}{1}) admits an E^{4} geometry.

Let

Nil^{3} = \{( \begin{matrix}1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix} ) | x, y, z ∈ ℝ\};

Then N il^{3} naturally has the structure of an ℝ−bundle over ℝ^{2}:

0 ⟶ ⟨z⟩ ⟶ ⟨x, y, z⟩ ⟶ ⟨x, y⟩ ⟶ 1.

This extension doesn’t split since [⟨x⟩, ⟨y⟩] = ⟨z⟩. All isometries of N il^{3} preserve orientation and the line bundle structure over ℝ^{2} so we have an induced exact sequence:

0 ⟶ I som^{+}(E) ⟶ I som(Nil^{3}) ⟶ I som(E^{2}) ⟶ 1.

Here I som^{+}(E) = ℝ is the group of orientation preserving isometries of E. The identity component I som^{0}(Nil^{3}) of I som(Nil^{3}) is N il^{3} ⋊ SO(2) where N il^{3} acts by left multiplication and the action of θ ∈ SO(2) is defined by:

(x, y, z)⟼^{ρ_{θ}}(cx + sy, −sx + cy, z + s(cy^{2} − cx^{2} − 2sxy)/2);

where s = sin(θ), c = cos(θ) and θ ∈ [0, 2π). We take I som(Nil^{3})/I som^{0}(Nil^{3}) to be generated by the automorphism (x, y, z)⟼^{λ}(x, −y, −z). Furthermore we have I som(Nil^{3}×E) = I som(Nil^{3})×I som(E) where I som(E) = ℝ⋊ℤ/2ℤ. The action of ℤ/2ℤ = ⟨𝔯⟩ on ℝ is by reflection or multiplication by -1. Below we shall denote the isometry of left multiplication by X ∈ N il^{3} simply by X.

Lemma. Let [figure: ℘: Isom(Nil^{3} × E); Isom(Nil^{3}); Isom(E^{2})] be the composite of the natural projection followed by the induced map of the ℝ-bundle projection. A representation Φ: Π_{1}(M) → I som(Nil^{3} × E) is faithful and discrete with cocompact image precisely when the following two properties are satisfied:

(1) The subgroup K ⊴ Π_{1}(M) whose image Φ(K) = Φ(Π_{1}(M)) ∩ ker(℘) embeds faithfully, discretely and cocompactly in ℝ × I som(E).

(2) The quotient group Π_{1}(M)/K embeds faithfully and discretely in I som(E^{2}) with cocompact image under the induced map from Φ.

proof: If Φ(Π_{1}(M)) is discrete in I som(Nil^{3}×E) then clearly Φ(K) is discrete in I som(Nil^{3}× E) and hence discrete in Isom^{+}(E). Hence by way of the quotient topology, Φ(Π_{1}(M))/Φ(K) must be discrete in Isom(E^{2}). Similarly if Π_{1}(M) maps faithfully into I som(Nil^{3} × E) then so too does K; in which case Π_{1}(M)/K must map faithfully into I som(E^{2}) via the induced map. If Φ(Π_{1}(M)) is cocompact then the fibration (Nil^{3} × E)/Φ(Π_{1}(M))⟶^{\overline{p}}E^{2}/\overline{Φ}(Π_{1}(M)/K) shows that \overline{Φ}(Π_{1}(M)/K) is cocompact. Moreover the fibres of \overline{p} are closed in (Nil^{3} × E)/Φ(Π_{1}(M)) and hence compact. Thus E^{2}/Φ(K) is compact. Below we prove the reverse implication. Let X = Ker(Φ) so that we have a commutative diagram as shown below:

[figure: K ∩ X; X; X/(K ∩ X); ψ; K; Φ|K; Π_{1}(M); Φ; Π_{1}(M)/K; \overline{Φ}; ℝ × I som(E); I som(Nil^{3} × E); I som(E^{2})]

By chasing a nontrivial element in K ∩ X into I som(Nil^{3} × E) we see that K ∩ X is trivial. In which case the map ψ is injective. Hence if x ∈ X is non-trivial then \overline{Φ}ψ(x) is non-trivial, contradicting the fact that x ∈ Ker(Φ). Thus X = \{1\} and Φ is faithful. If both Φ(K) ⊴ Isom^{+}(E) × Isom(E) and \overline{Φ}(Π_{1}(M)/K) ⊴ Isom(E^{2}) are discrete. Then Φ(Π_{1}(M)) ⊴ I som(Nil^{3} ×E) is discrete by way of the quotient topology. If Φ(K) is cocompact then we have a covering space projection I som(Nil^{3} × E)/Φ(K)⟶^{p}I som(Nil^{3} × E)/Φ(Π_{1}(M)) showing that Φ(Π_{1}(M)) is cocompact. □

The family of bundle-spaces (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}0 \\ n \end{matrix} ))_{n∈ℕ\\{0\}} admit geometries of

type Nil^{3} × E.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h), \\ [α, β] = h^{n}, & β(l, h)β^{−1} = (l, h). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(l)(x, y, z, w) = (x, y, z, w + 1),
i.e Φ(l) = (I, (1, 1)).
Φ(h)(x, y, z, w) = (x, y, z + h_{0}, w)
i.e Φ(h) = (( \begin{matrix}1 & 0 & h_{0} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(β)(x, y, z, w) = (x, y + 1, z, w)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(α)(x, y, z, w) = (x + 1, y, z + y, −w)
i.e Φ(α) = (( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯)).

where h_{0} is non-zero.

The relations [l, h] = 1, β(l, h)β^{−1} = (l, h) and α(l, h)α^{−1} = (l^{−1}, h) always hold. The relation [α, β] = h^{n} gives h_{0} = 1/n and we have the desired representation. To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(l), Φ(h)⟩ ≅ ℤ^{2}. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient

T^{2}. Similarly \overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)↦^{[α]}(x + 1, y), (x, y)↦^{[β]}(x, y + 1)⟩ ≅ ℤ^{2} acts on ℝ^{2}⟨x, y⟩ with quotient T^{2} so we have the desired representation.

The family of bundle-spaces (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}1 \\ n \end{matrix} ))_{n∈ℕ\\{0\}} admit geometries of

type Nil^{3} × E.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h), \\ [α, β] = lh^{n}, & β(l, h)β^{−1} = (l, h). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(l)(x, y, z, w) = (x, y, z, w + 1),
i.e Φ(l) = (I, (1, 1)).
Φ(h)(x, y, z, w) = (x, y, z + 1, w)
i.e Φ(h) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(β)(x, y, z, w) = (x, y + 1, z, w + k)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , (k, 1)).
Φ(α)(x, y, z, w) = (x + a, y, z + ay, −w)
i.e Φ(α) = (( \begin{matrix}1 & a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯)).

The relations [l, h] = 1, α(l, h)α^{−1} = (l^{−1}, h) and β(l, h)β^{−1} = (l, h) are automatically satisfied. The relation [α, β] = lh^{n} gives a = n and k = −1/2. To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(l), Φ(h)⟩ ≅ ℤ^{2}. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient

T^{2}. Similarly \overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)↦^{[α]}(x + n, y), (x, y)↦^{[β]}(x, y + 1)⟩ ≅ ℤ^{2} acts on ℝ^{2}⟨x, y⟩ with quotient T^{2} so we have the desired representation.

The family of bundle-spaces (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}0 \\ n \end{matrix} ))_{n∈ℕ\\{0\}} admit geometries of

type Nil^{3} × E.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, lh), \\ [α, β] = h^{n}, & β(l, h)β^{−1} = (l, h). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(l)(x, y, z, w) = (x, y, z, w + 1),
i.e Φ(l) = (I, (1, 1)).
Φ(h)(x, y, z, w) = (x, y, z + 1, w + m)
i.e Φ(h) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (m, 1)).
Φ(β)(x, y, z, w) = (x, y + 1, z, w + k)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , (k, 1)).
Φ(α)(x, y, z, w) = (x + a, y, z + ay, −w)
i.e Φ(α) = (( \begin{matrix}1 & a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯)).

The relations [l, h] = 1, β(l, h)β^{−1} = (l, h) and αlα^{−1} = l^{−1} always hold. The relation αhα^{−1} = lh gives m = −1/2. The relation [α, β] = h^{n} gives a = n and k = n/4. To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(l), Φ(h)⟩ ≅ ℤ^{2}. Clearly K acts on ℝ^{2}⟨z, w⟩ with

quotient T^{2}. Similarly \overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)↦^{[α]}(x + n, y), (x, y)↦^{[β]}(x, y + 1)⟩ ≅ ℤ^{2} acts on ℝ^{2}⟨x, y⟩ with quotient T^{2} so we have the desired representation.

The Lie group Sol^{3} = ℝ^{2}⋊ℝ where the action of t ∈ ℝ on ℝ^{2} = \{( \begin{matrix}x \\ y \end{matrix} )\} is left mul-

tiplication by the matrix ({e^{t} \atop 0} {0 \atop e^{−t}} ). The surfaces defined by constant values of t foli-

ate Sol^{3} and are preserved by all isometries. The identity component I som^{0}(Sol^{3}) of the isometry group is again Sol^{3}. The quotient group I som(Sol^{3})/I som^{0}(Sol^{3}) is isomorphic to the dihedral group of order 8 generated by the automorphisms:

(x, y, t) ⟼ (±x, ±y, t), and (x, y, t) ⟼ (±y, ±x, −t).

As before we have I som(Sol^{3} × E) = I som(Sol^{3}) × I som(E) so that we have an induced exact sequence:

1 ⟶ Isom(E^{2}) ⟶ Isom(Sol^{3} × E) ⟶ Isom(E) × Isom(E) ⟶ 1.

Where Isom(E) × Isom(E) acts on ℝ^{2} as a subgroup of Isom(E^{2}).

Lemma. Let [figure: ℘: Isom(Sol^{3} × E); Isom(E) × Isom(E)] be the above projec-

tion. A representation Φ: Π_{1}(M) → I som(Sol^{3} × E) is faithful and discrete with cocompact image precisely when:

(1) The subgroup K ⊴ Π_{1}(M) whose image Φ(K) = Φ(Π_{1}(M))∩ker(℘) embeds faithfully, discretely and cocompactly in Isom(E^{2}).

(2) The quotient group Π_{1}(M)/K embeds faithfully and discretely in Isom(E)× Isom(E) with cocompact image under the induced map from Φ.

proof: If Φ(Π_{1}(M)) maps discretely into Isom(Sol^{3} ×E) then so too does Φ(K) and hence is discrete in Isom(E^{2}). Hence by way of the quotient topology Φ(Π_{1}(M))/Φ(K) is discrete in Isom(E) × Isom(E). Similarly if Φ(Π_{1}(M)) maps faithfully into Isom(Sol^{3} × E) then so does Φ(K); in which case Π_{1}(M)/K must map faithfully into Isom(E) × Isom(E) via the induced map. If Φ(Π_{1}(M)) is cocompact then the fibration (Sol^{3} × E)/Φ(Π_{1}(M))⟶^{\overline{p}}E^{2}/\overline{Φ}(Π_{1}(M)/K) shows that \overline{Φ}(Π_{1}(M)/K) is cocompact. Moreover the fibres of \overline{p} are closed in (Sol^{3} × E)/Φ(Π_{1}(M)) and hence compact. Thus E^{2}/Φ(K) is compact. Below we prove the reverse implication. Let X = Ker(Φ) so that we have a commutative diagram as shown below:

[figure: K ∩ X; X; X/(K ∩ X); ψ; K; Φ|K; Π_{1}(M); Φ; Π_{1}(M)/K; \overline{Φ}; I som(E^{2}); I som(Sol^{3} × E); Isom(E) × Isom(E)]

By chasing a nontrivial element in K ∩ X into I som(Sol^{3} × E) we see that K ∩ X is trivial. In which case the map ψ is injective. Hence if x ∈ X is non-trivial then \overline{Φ}ψ(x) is non-trivial contradicting the fact that x ∈ Ker(Φ). Thus X = \{1\} and Φ is faithful. If both Φ(K) ⊴ Isom(E^{2}) and \overline{Φ}(Π_{1}(M)/K) ⊴ Isom(E) × Isom(E) are discrete. Then Φ(Π_{1}(M)) ⊴ I som(Sol^{3} × E) is discrete by way of the quotient topology. If Φ(K) is cocompact then we have a covering space projection I som(Sol^{3} × E)/Φ(K)⟶^{p}I som(Sol^{3} × E)/Φ(Π_{1}(M)) showing that Φ(Π_{1}(M)) is cocompact. □

The family of bundle-spaces (A, ±I; ( \begin{matrix}a \\ b \end{matrix} ), det(A) = −1, rank(A−I) = 2) admit geometries of type Sol^{3} × E.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h)A, \\ [α, β] = l^{a}h^{b}, & β(l, h)β^{−1} = (l^{±1}, h^{±1}). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Sol^{3} × E) as follows: We have that P AP^{−1} = ({e_{0}^{t} \atop 0} {0 \atop −e^{−t_{0}}} ) for some P = ({p \atop r} {q \atop s}) ∈ GL(2, ℝ). Hence we define a

representation Φ: Π_{1}(M) → I som(Sol^{3} × E) as follows:

Φ(l)(x, y, t, u) = (x + p, y + r, t, u)
ie Φ(l) = ((\binom{p}{q}, 0), (0, 1)).
Φ(h)(x, y, t, u) = (x + q, y + s, t, u)
ie Φ(h) = ((\binom{r}{s}, 0), (0, 1)).
Φ(β)(x, y, t, u) = (ϵx + x_{1}, ϵy + y_{1}, t, u + 1)
ie Φ(β) = ((\binom{x_{1}}{y_{1}}, 0), (1, 1))λ.
Φ(α)(x, y, t, u) = (e^{t_{0}}x + x_{0}, −e^{−t_{0}}y + y_{0}, t + t_{0}, u)
ie Φ(α) = ((\binom{x_{0}}{y_{0}}, t_{0}), (0, 1))ρ t_{0} ≠ 0.

where ϵ = ±1 respective as to whether B = ±I and hence (x, y, t, u)⟼^{λ}(ϵx, ϵy, t, u). Similarly ρ is the isometry (x, y, t, u)⟼^{ρ}(x, −y, t, u).

The relations [l, h] = 1, α(l, h)α^{−1} = (l, h)A and β(l, h)β^{−1} = (l^{±1}, h^{±1}) are automatically satisfied. The relation [α, β] = l^{a}h^{b} gives the relation:

\binom{x_{1}}{y_{1}} = (I − P AP^{−1})^{−1} ((1 − ϵ)\binom{x_{0}}{y_{0}} − P \binom{a}{b}) .

Here K = ⟨Φ(l), Φ(h)⟩. Since P ∈ GL(2, ℝ) the vectors ( \begin{matrix}p \\ r \end{matrix} ) and ( \begin{matrix}q \\ s \end{matrix} ) are linearly independent and hence K acts on ℝ^{2}⟨x, y⟩ with quotient T^{2}. Similarly the group

\overline{Φ}(Π_{1}(M)/K) = ⟨t↦^{[α]}t + t_{0}, u↦^{[β]}t + 1⟩ is isomorphic to ℤ^{2} and acts on ℝ^{2}⟨t, u⟩ with qutient T^{2}. Thus Φ is indeed faithful and discrete with cocompact image; and we have the desired representation completing the proof. □

The Classification of Non-Orientable Total Spaces of T^{2}-Bundles over Kb and their Geometries

Theorem. Let M be the non-orientable total space of a T^{2}-bundle over Kb with first Betti number β_{1}(M). Then M is bundle diffeomorphic to one of the geometric 4-manifolds listed below:

Total Space  Geometric Type

β_{1}(M) = 3  (I, I; ({(1) \atop 0}))  E^{4}

β_{1}(M) = 2  \begin{matrix}(( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , I; ({(1) \atop (1)}))_{λ∈ℕ\\{0\}} & Nil^{3} × E \\ (I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) ; ({(1) \atop (1)})) & E^{4} \\ (I, ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) ; ( \begin{matrix}0 \\ (1) \end{matrix} )) & E^{4} \end{matrix}

β_{1}(M) = 1

(−I, I; ( \begin{matrix}0 \\ 0 \end{matrix} ))  E^{4}
(−I, I; ( \begin{matrix}1 \\ 0 \end{matrix} ))  Nil^{3} × E
(−I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) ; ( \begin{matrix}(1) \\ 0 \end{matrix} ))  E^{4}
(−I, ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) ; ( \begin{matrix}0 \\ n \end{matrix}))_{n∈ℕ\\{0\}}  Nil^{3} × E
(( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ) , I; ( \begin{matrix}m \\ n \end{matrix}))_{λ,n≠0}  Nil^{4}
(I, −I; ( \begin{matrix}0 \\ 0 \end{matrix} ))  E^{4}
(I, −I; ( \begin{matrix}1 \\ 0 \end{matrix} ))  Nil^{3} × E
(( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , −I; ({(1) \atop (1)}))_{λ≠0}  Nil^{3} × E
(A, ±I; ( \begin{matrix}a \\ b \end{matrix}), det(A) = 1, |trace(A)| > 2)  Sol^{3} × E
(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , B; ( \begin{matrix}a \\ b \end{matrix}), det(B) = 1, |trace(B)| > 2)  Sol^{3} × E
(−I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) ; ( \begin{matrix}(1) \\ n \end{matrix}))_{n∈ℕ\\{0\}}  Nil^{3} × E

Table 3.

Remark. As before the classes of Sol^{3} × E-manifolds listed above admit further refinement, see [Ne].

By writing M = (A, B; ( \begin{matrix}a \\ b \end{matrix} )) we are refering to the bundle-space with fundamental group

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h)A, \\ αβα^{−1}β = l^{a}h^{b}, & β(l, h)β^{−1} = (l, h)B. \end{matrix} ⟩ .

The following three classes of fibre preserving diffeomorphims correspond to the respective transformations of the algebraic representation (A, B; \binom{a}{b}) of the bundle space M:

(1) Change of framing of the fibre

(l, h) ↦ (l, h)P, ∃P ∈ GL(2, ℤ).
(A, B; ( \begin{matrix}a \\ b \end{matrix} )) ↦ (P^{−1}AP, P^{−1}BP; P^{−1}( \begin{matrix}a \\ b \end{matrix} )).

(2) Change of base of the base space

(i)  (α, β) ↦ (α^{−1}, β)
(A, B; ( \begin{matrix}a \\ b \end{matrix} )) ↦ (A^{−1}, B; B^{−1}A^{−1}( \begin{matrix}a \\ b \end{matrix} )).
(ii)  (α, β) ↦ (α, β^{−1})
(A, B; ( \begin{matrix}a \\ b \end{matrix} )) ↦ (A, B^{−1}; −B( \begin{matrix}a \\ b \end{matrix} )).
(iii)  (α, β) ↦ (βαβ^{−1}, β)
(A, B; ( \begin{matrix}a \\ b \end{matrix} )) ↦ (BAB^{−1}, B; B( \begin{matrix}a \\ b \end{matrix} )).
(iv)  (α, β) ↦ (α^{−1}β, α^{−1}βα)
(A, B; ( \begin{matrix}a \\ b \end{matrix} )) ↦ (A^{−1}B, B^{−1}; A^{−2}( \begin{matrix}a \\ b \end{matrix} )).

(3) Change of cross-section

(α, β) ↦ (αl^{p}h^{q}, βl^{s}h^{t}), where t, s, p, q ∈ ℤ are free.
(A, B; ( \begin{matrix}a \\ b \end{matrix} )) ↦ (A, B; ( \begin{matrix}a \\ b \end{matrix} ) + B^{−1}A(B^{−1} − I)( \begin{matrix}p \\ q \end{matrix}) + (B^{−1}A + I) ( \begin{matrix}s \\ t \end{matrix} )).

Here equivalence classes of triples modulo transformations of type (3) are in bijection with the elements of H^{2}(ℤ˜×ℤ; ℤ^{2}) ≅ ℤ^{2}/⟨−A − I, B − I⟩.

The bundle-space M is orientable precisely when det(A) = −1 and det(B) = 1. Below we shall only consider the two cases: det(A) = det(B). For if det(A) = 1 and det(B) = −1 we may replace A by A^{−1}B. On examination of the presentation for H_{1}(M) we see that β_{1}(M) = 3 − rank(A − I, B − I).

Assuming that the bundle-spaces listed in table 3. admit the geometries as claimed, then to prove that table 3. is a complete listing of all such distinct bundlespaces up fibre preserving diffeomorphism it suffices to distinguish between representations admitting identical geometries with the same values of β_{1}(M).

Lemma 2. Let the matrix A ∈ GL(2, ℤ) be such that A^{2} = I.

(1) If det(A) = 1 then A is conjugate to either ±I.

(2) If det(A) = −1 then A is conjugate to one of ({−1 \atop 0} {(1) \atop 1}).

proof : The eigenvalues of A square to give 1 and therefore belong to the set {±1}. Hence the result follows from Lemma 1. □

A) The case β_{1}(M) = 3 ⟹ rank(A − I, B − I) = 0. Here we must have A = I = B and hence by an appropriate change of cross-section we may assume ( \begin{matrix}a \\ b \end{matrix} ) = \binom{(1)}{(1)}; corresponding to H^{2}(ℤ˜×ℤ; ℤ^{2}) ≅ ℤ/2ℤ ⊕ ℤ/2ℤ, being the four extension equivalence classes with trivial action. However by changing the framing of the fibre we see that there are at most two such bundle-spaces distinguished by whether ( \begin{matrix}a \\ b \end{matrix} ) = \binom{0}{0} or not:

(I, I; ( \begin{matrix}0 \\ 0 \end{matrix} )), (I, I; ( \begin{matrix}1 \\ 0 \end{matrix} )).

These two manifolds are indeed distinct as their first homology groups are ℤ^{3}⊕ℤ/2ℤ and ℤ^{3} respectively. The first manifold is the product bundle Kb × T^{2}.

B) The case β_{1}(M) = 2 ⟹ rank(A − I, B − I) = 1. B_{1}) The case det(A) = 1 = det(B)

B_{11}) rank(A − I) = 0 ⟹ A = I.

We must have rank(B −I) = 1 therefore B ∼ ({1 \atop 0} {λ \atop 1})_{λ≠0}. However the relation

ABA^{−1}B = I implies B^{2} = I and consequently λ = 0 so we have no such bundle-spaces.

B_{12}) rank(A − I) = 1 ⟹ A ∼ ({1 \atop 0} {λ \atop 1})_{λ≠0}

Furthermore by conjugating A by ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) if nessesary, we may assume that λ

is positive. If rank(B−I) = 0 then B = I and we may transform ( \begin{matrix}a \\ b \end{matrix} ) into \binom{(1)}{(1)} if λ is even, or \binom{0}{(1)} if λ is odd by some appropriate change of cross-section. This corresponds to the fact that

H^{2}(ℤ˜×ℤ; ℤ) ≅ \{ \begin{matrix}ℤ/2ℤ ⊕ ℤ/2ℤ, & if λ ∈ 2ℕ; \\ ℤ/2ℤ, & if λ ∉ 2ℕ. \end{matrix}

That is for each value of λ ∈ ℕ we obtain two triples when λ is odd, and four triples when λ is even:

(( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}(1) \\ (1) \end{matrix} ))_{λ∈2ℕ\\{0\}}, (( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}0 \\ (1) \end{matrix} ))_{λ∈ℕ\2ℕ}.

However when λ is even, a tranformation of the fibre via the matrix P = ({1 \atop 0} {1 \atop 1}) equates the two representations with Euler numbers ({1 \atop 0}) and ({1 \atop 1}). For the sake of brevity we tabulate this redundancy as it is shown above. Abelianization of these groups yields: ℤ^{2} ⊕ ℤ/2ℤ ⊕ ℤ/λℤ or ℤ^{2} ⊕ ℤ/2λℤ when ( \begin{matrix}a \\ b \end{matrix} ) = \binom{0}{0} or \binom{0}{1} respectively. And the remaining bundle-space has H_{1}(M) ≅ ℤ^{2} ⊕ ℤ/λℤ; showing that these five infinite families of manifolds are all mutually nondiffeomorphic.

If rank(B−I) = 1 we must have B ∼ ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} )_{μ≠0}. However the relation ABA^{−1}B =

I implies that μ = 0 and we have no such bundle-spaces. B_{2}) The case det(A) = −1 = det(B) ⟹ rank(A − I) = 1 = rank(B − I).

Here we may assume A = ({−1 \atop 0} {(1) \atop 1}) in which case we must have B = ( \begin{matrix}−1 & ∗ \\ 0 & 1 \end{matrix} ). The

relation ABA^{−1}B = I gives ∗ = (1) so that after some changes of cross-section we

obtain the six representations: (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}(1) \\ (1) \end{matrix} )),and (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ (1) \end{matrix} )).

In this case for cosmetic purposes we abandon our convention that det(A) = det(B) and replace A by A^{−1}B so as to write these representations as:

(I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}(1) \\ (1) \end{matrix} )), (I, ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ (1) \end{matrix} )).

These triples represent the equivalence classes of the associated extensions since

H^{2}(ℤ˜×ℤ; ℤ) ≅ \{ \begin{matrix}ℤ/2ℤ ⊕ ℤ/2ℤ, & if B = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); \\ ℤ/2ℤ, & if B = ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ). \end{matrix}

When B = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) the groups H_{1}(M) are: ℤ^{2} ⊕ (ℤ/2ℤ)^{2}, ℤ^{2} ⊕ ℤ/4ℤ, ℤ^{2} ⊕ ℤ/2ℤ

or ℤ^{2} ⊕ ℤ/2ℤ depending on whether ( \begin{matrix}a \\ b \end{matrix} ) = \binom{0}{0}, \binom{1}{0}, \binom{0}{1} or \binom{1}{1} respectively. These latter two representations with isomorphic first homology are seen to be distinct by observing the isomorphism type of the quotient of Π_{1}(M) by its center Z(Π_{1}(M)) = ⟨α^{2}, h⟩. Since the center is characteristic in Π_{1}(M) any isomorphism of Π_{1}(M) will induce an isomorphism on this quotient. When ( \begin{matrix}a \\ b \end{matrix} ) = \binom{0}{1} this quotient abelianizes to give (ℤ/2ℤ)^{3} and ℤ/2ℤ ⊕ ℤ/4ℤ in the

other case. Thus these four bundle-spaces are distinct. When B = ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) the

group H_{1}(M) is either ℤ^{2} ⊕ ℤ/2ℤ or ℤ^{2} depending on whether ( \begin{matrix}a \\ b \end{matrix} ) = \binom{0}{0} or \binom{0}{1} respectively. The former case may be distinguished from the other two bundlespaces with the same first homology group as follows. The kernel K ◁ Π_{1}(M) of the projection:

[figure: Π_{1}(M); H_{1}(M); H_{1}(M)/Torsion ≅ ℤ/2ℤ.]

is characterstic and is isomorphic to the Klein bottle group generated by β and l. However the above subgroup K = ⟨l, β^{2}h^{−1}⟩ of the bundle-spaces (I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}(1) \\ 1 \end{matrix} )) is free abelian.

C) The case β_{1}(M) = 1 ⟹ rank(A − I, B − I) = 2. Here we shall utilize the fact that M is double covered by the total-space M̃ of a T^{2}-bundle over T^{2}. The covering projection [figure: M̃:; M] is induced by the orientation double covering [figure: T^{2}; Kb] and hence is associated with the kernel of the composite projection:

[figure: ð: Π_{1}(M); H_{1}(M); H_{1}(M)/Torsion; ℤ/2ℤ.]

Thus M̃ is the bundle-space with algebraic representation (A^{2}, B; (A − B)( \begin{matrix}a \\ b \end{matrix} )). Since Ker(ð) is characteristic in Π_{1}(M) any isomorphism ψ: Π_{1}(M) → Π_{1}(M′) induces an isomorphism ψ̃: Π_{1}(M̃) → Π_{1}(M̃′). However the same bundle-space M̃ may cover many distinct T^{2}-bundle spaces over Kb. We proceed in cases according to the value of β_{1}(M̃).

C_{1}) The case β_{1}(M̃) = 4 ⟹ M̃ = T^{4} We must have A^{2} = I = B, thus det(A) = 1 and hence by Lemma 2. A = ±I. However A = I contradicts rank(A − I, B − I) = 2 so we must have A = −I. Therefore the matrix A − B is invertible over ℝ and thus the Euler number is trivial. That is we obtain the sole representaion:

(−I, I; ( \begin{matrix}0 \\ 0 \end{matrix} )).

Here H_{1}(M) ≃ ℤ ⊕ (ℤ/2ℤ)^{3}. C_{2}) The case β_{1}(M̃) = 3 ⟹ rank(A^{2} − I, B − I; (A − B)( \begin{matrix}a \\ b \end{matrix} )) = 1.

C_{21}) The case det(A) = 1 = det(B). Here we have det(A^{2}) = 1 = det(B) so that M̃ is orientable. From Table 2. we see that there is only one orientable T^{2}-bundle over T^{2}. Therefore A^{2} = I = B so that from Lemma 2 we must have A = ±I. As in the previous case A = I yields a contradiction so A = −I. Thus we obtain a bigraded infinite family of triples indexed by ( \begin{matrix}a \\ b \end{matrix} ), in correspondence with H^{2}(ℤ˜×ℤ; ℤ^{2}) ≅ ℤ^{2}, representing extension equivalence classes with the given action. [note: algebraic transformations of type (3) for such an action act identically on ℤ^{2}.] However as both ±I are invariant to a change of basis we obtain just two representations up to extension isomorphism distinguished by whether ( \begin{matrix}a \\ b \end{matrix} ) is the zero vector or not. As one of these representations was already discovered in the previous case, we just obtain the single representation:

(−I, I; ( \begin{matrix}1 \\ 0 \end{matrix} ));

with H_{1}(M) ≃ ℤ ⊕ ℤ/2ℤ ⊕ ℤ/4ℤ. C_{22}) The case det(A) = −1 = det(B).

Here det(A^{2}) = 1 and rank(A^{2} − I) ≤ 1, thus A^{2} ∼ ({1 \atop 0} {λ \atop 1}) for some λ ∈

ℤ. Writing A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and examining this relation with the condition that det(A) = −1, we find the only posibility is that λ = 0. Hence by lemma 2 we may assume A = ({−1 \atop 0} {(1) \atop 1}).

If (1)=0 then writing B as ( \begin{matrix}p & q \\ r & s \end{matrix} ) and testing the relation ABA^{−1}B = I with the condition that det(B) = −1 we see that the only possibilities for B are ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) or ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). However the latter matrix contradicts the assumption

that rank(A − I, B − I) = 2 so we must have B = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). Now a change of

cross-section permits us to reduce the Euler number to ( \begin{matrix}a \\ (1) \end{matrix} ) corresponding to

H^{2}(ℤ˜×ℤ; ℤ^{2}) ≅ ℤ⊕ℤ/2ℤ. However the condition that rank(A^{2}−I, B−I; (A− B)( \begin{matrix}a \\ b \end{matrix} )) = 1 implies that a = 0 so that we obtain the two representations:

(( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ); ( \begin{matrix}0 \\ (1) \end{matrix} )). We now choose to cast aside our convention and write these representations as:

(−I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 0 \end{matrix} )), (−I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}1 \\ 0 \end{matrix} )).

These two representations have H_{1}(M) = ℤ ⊕ (ℤ/2ℤ)^{3} or ℤ ⊕ ℤ/4ℤ ⊕ ℤ/2ℤ depending on whether ( \begin{matrix}a \\ b \end{matrix} ) = ({0 \atop 0}) or ({0 \atop 1}) respectively; and hence represent distinct manifolds.

If the entry (1) in A is non-trivial then following similar methods to the above we find that B = ( \begin{matrix}1 & −1 \\ 0 & −1 \end{matrix} ) and the set \{( \begin{matrix}a \\ b \end{matrix} ), \binom{1}{2}\} is ℤ-linearly indepen-

dent. Now conjugating A and B by ± ( \begin{matrix}−1 & 0 \\ −2 & −1 \end{matrix} ) and then replacing A by A^{−1}B and applying a cross-section change we obtain the family of representations:

(−I, ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ n \end{matrix} ))_{n∈ℕ\\{0\}.}

Being in correspondence with equivalence classes given by H^{2}(ℤ˜×ℤ; ℤ^{2}) ≅ ℤ. Here H_{1}(M) = ℤ ⊕ (ℤ/2ℤ)^{2} distinguishing this family of bundle-spaces from the other Nil^{3} ×E-manifold with β_{1}(M̃) = 3 [case C_{21})]. For fixed n ∈ ℕ such a bundle-space is doubly covered by the T^{2}-bundle over T^{2} with characteristic triple: (( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ) , I; \binom{0}{2n}). The diffeomorphism type of such a bundle-space is determined by n, and hence the above family of triples represent an infinite mutually non-diffeomorphic family of T^{2}-bundle-spaces over Kb indexed by n ∈ ℕ.

C_{3}) The case β_{1}(M̃) = 2 ⟹ rank(A^{2} − I, B − I; (A − B)( \begin{matrix}a \\ b \end{matrix} )) = 2.

C_{31}) The case det(A) = 1 = det(B).

If rank(B − I) = 0 then B = I and hence A must be one of the matrices: I, ({1 \atop 0} {λ \atop 1}) , ( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ) or else A has irrational eigenvalues. The first two matrices contradict rank(A − I, B − I) = 2 so do not yield a representation. Therefore A = ( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ) and H^{2}(ℤ˜×ℤ; ℤ^{2}) ≅ ℤ ⊕ ℤ/λℤ corresponding to the fact that an appropriate change of cross-section assures that b is positive and reduces a modulo λ. Furthermore the condition rank(A^{2} − I, B − I; (A − B)( \begin{matrix}a \\ b \end{matrix} )) = 2 shows that b ≠ 0. That is we obtain the two families of representations:

(( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ) , I; ( \begin{matrix}m \\ n \end{matrix} ))_{n≠0}^{0≤m<λ}, (A, I; ( \begin{matrix}a \\ b \end{matrix} ), det(A) = 1, |trace(A)| > 2).

A representative set of distinct bundle-spaces represented by the above triples is in bijection with the module ℤ^{2}/⟨−A−I, A^{−2} −I, P ⟩ where the matrix P ∈ GL(2, ℤ) ranges over the centralizer of A in GL(2, ℤ). If rank(B − I) = 1 then we may assume B = ({1 \atop 0} {λ \atop 1})_{λ≠0}. However an easy calculation shows that

there are no matrices A ∈ SL(2, ℤ) that satisfy AB = B^{−1}A. Thus we get no manifolds under these conditions.

If rank(B − I) = 2 then we may assume exclusively either B = ( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ) or B has irrational eigenvalues. In the first case examining the relation AB = B^{−1}A with the condition that det(A) = 1 shows that λ = 0. That is B = −I and the possiblities for A are: I, ( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ) or that A has irrational eigenvalues.

In the first instance, a change of cross-section reduces the Euler number to ({(1) \atop (1)}) in correspondence with H^{2}(ℤ˜×ℤ; ℤ^{2}) ≅ ℤ/2ℤ ⊕ ℤ/2ℤ. However by changing the framing of the fibre we see that there are just two distinct representations depending on whether ( \begin{matrix}a \\ b \end{matrix} ) is trivial or not. That is we obtain the pair of triples:

(I, −I; ( \begin{matrix}0 \\ 0 \end{matrix} )), (I, −I; ( \begin{matrix}1 \\ 0 \end{matrix} )).

These triples represent non-diffeomorphic manifolds which are distinguished by H_{1}(M) being isomorphic to ℤ ⊕ (ℤ/2ℤ)^{3} or ℤ ⊕ ℤ/2ℤ ⊕ ℤ/4ℤ respectively.

By replacing A by A^{−1}B we see that the following two possibilities for A give rise to the same family of representations [indexed by λ]. As before, modulo a change in the fibre we may assume that λ ∈ ℕ. Furthermore we are able transform the cross-section for this family so that ( \begin{matrix}a \\ b \end{matrix} ) = \binom{(1)}{(1)} if λ is even, or \binom{0}{(1)} if λ is odd; in accordance with H^{2}(ℤ˜×ℤ; ℤ^{2}) having order four or two respectively. However when λ is even, the Euler number admits additional refinement via a change in the fibre equating ({1 \atop 1}) and ({1 \atop 0}). That is we obtain the six infinite families of representations:

(( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}a \\ b \end{matrix} ))_{λ∈2ℕ}, (( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}0 \\ (1) \end{matrix} ))_{λ∈ℕ\2ℕ};

where ( \begin{matrix}a \\ b \end{matrix} ) belongs to the set \{({0 \atop 0}), ({1 \atop 0}), ({0 \atop 1})\}. These triples are seen to represent distinct bundle-spaces as follows.

If λ is even, then the triple with trivial Euler number has H_{1}(M) ≅ ℤ ⊕ (ℤ/2ℤ)^{2} in contrast to the other two families with H_{1}(M) ≅ ℤ⊕ℤ/2ℤ⊕ℤ/4ℤ. These remaining two bundle-spaces are distinguished by their double covers.

If λ is odd then H_{1}(M) ≅ ℤ ⊕ (ℤ/2ℤ)^{2} when the Euler number is trivial and is isomorphic to ℤ ⊕ ℤ/4ℤ otherwise.

Finally when A has irrational eigenvalues we obtain the family of representations:

(A, −I; ( \begin{matrix}a \\ b \end{matrix} ), det(A) = 1, |trace(A)| > 2).

Note, there are at most four bundle-spaces corresponding to each choice of A since H^{2}(ℤ˜×ℤ; ℤ^{2}) is some quotient of ℤ/2ℤ ⊕ ℤ/2ℤ. Here the double covering of (I, −I; ( \begin{matrix}1 \\ 0 \end{matrix} )) is (I, −I; ( \begin{matrix}−2 \\ 0 \end{matrix} )) in contrast to the double coverings

of the manifolds (({1 \atop 0} {λ \atop 1}) , −I; ( \begin{matrix}(1) \\ (1) \end{matrix} )) and (( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ) , −I; ( \begin{matrix}(1) \\ (1) \end{matrix} )) which are of

the form (( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}a \\ b \end{matrix} ))_{μ≠0}: a distinct family of bundle-spaces to the above

Nil^{3} ×E-manifold [from table 2. ]. This latter family (A, −I; ( \begin{matrix}a \\ b \end{matrix} ), det(A) = 1, |trace(A)| > 2) of bundle-spaces differs from the previous family of Sol^{3} ×E bundle-spaces discovered; since all algebraic transformations preserve the conjugacy class of B. For the sake of brevity we tabulate these two families together. If B has irrational eigenvalues then there exists a matrix

P ∈ GL(2, ℝ) such that P^{−1}BP = ( \begin{matrix}e^{t} & 0 \\ 0 & −e^{t} \end{matrix} )_{t≠0}. Writing A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and

examining the relation P^{−1}ABA^{−1}BP = I we see that A is ℝ-conjugate to ( \begin{matrix}0 & −s \\ s^{−1} & 0 \end{matrix} ) for some real s ≠ 0. Now conjugating this form of A by ( \begin{matrix}s^{−1/2} & 0 \\ 0 & s^{1/2} \end{matrix} )

we see that A is ℝ-conjugate to ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and hence ℤ-conjugate to the same matrix. That is we obtain the family of representations:

(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , B; ( \begin{matrix}a \\ b \end{matrix} ), det(B) = 1, |trace(B)| > 2).

[Note: for the above family of triples there are at most two distinct bundlespaces corresponding to each choice of B since H^{2}(ℤ˜×ℤ; ℤ^{2}) is some quotient of ℤ/2ℤ. Once again this family of Sol^{3} × E bundle-spaces differs from the previous two discovered as any algebraic transformation of this representation preserves the conjugacy class of the matrix B.

C_{32}) The case det(A) = −1 = det(B). In this case rank(B − I) ≥ 1. We proceed acordingly. If rank(B − I) = 1 then we may assume that B = ({−1 \atop 0} {(1) \atop 1}). If (1)=0 the relation AB = B^{−1}A

implies that A is diagonal; hence A = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) or ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) the former matrix

contradicts the assumtion that rank(A−I, B −I) = 2. Thus A = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) and

the condition rank(A^{2} − I, B − I; (A − B)( \begin{matrix}a \\ b \end{matrix} )) = 2 implies b ≠ 0. Hence on replacing A by A^{−1}B, then conjugating and applying a change of cross-section we obtain the family of representations:

(−I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}(1) \\ n \end{matrix} ))_{n∈ℕ\\{0\}.}

This family of triples represents the equivalence classes of the corresponding extensions with the above given action as H^{2}(ℤ˜×ℤ; ℤ^{2}) ≅ ℤ ⊕ ℤ/2ℤ. For a fixed n ∈ ℕ such bundle-spaces are doubly covered by the non-orientable T^{2}-bundle over T^{2} represented by the triple: (( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , I; \binom{0}{−2n}), showing that n is an invariant of the diffeomorphism type. To see that triples with Euler number differing in the first component represent distinct bundle-spaces, we consider the characteristic subgroup G ◁ Π_{1}(M) that is the intersection of the kernels of all homomorphsims κ: Π_{1}(M) → ℤ/2ℤ. Here we have:

G = \{ \begin{matrix}⟨α^{2}, β^{2}, l^{2}, h⟩, & if (1) = 0; \\ ⟨α^{2}, β^{2}, l^{2}, h^{2}⟩, & if (1) = 1; \end{matrix}

and hence in both cases the commutator subgroup of G,

[G, G] = ⟨α^{2}, β^{2}⟩ = ⟨h^{4n}⟩.

Thus the isolator I([G, G]) = ⟨h⟩ is characteristic in Π_{1}(M); and the respective abelianized quotients of Π_{1}(M) by ⟨h⟩ yield ℤ ⊕ (ℤ/2ℤ)^{2} or ℤ ⊕ ℤ/4ℤ, distinguishing the triples. Here H_{1}(M) = ℤ ⊕ (ℤ/2ℤ)^{3} when n is even and ℤ⊕ℤ/4ℤ⊕ℤ/2ℤ when n is odd when (1)=0. And H_{1}(M) = ℤ⊕ℤ/4ℤ⊕ℤ/2ℤ when (1)=1.

If the entry (1)=1 in B then following similar methods to above we find that A = ( \begin{matrix}1 & −1 \\ 0 & −1 \end{matrix} ) and b ≠ 0. Hence replacing A by A^{−1}B and changing the cross-section

we may write this family of representations as: (−I, ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ n \end{matrix} ))_{n∈ℕ\\{0\}}; which were already discovered in case C_{22}. If rank(B − I) = 2 then there exists a matrix P ∈ GL(2, ℝ) such that P^{−1}BP = ( \begin{matrix}e^{t} & 0 \\ 0 & −e^{−t} \end{matrix} )_{t≠0}. Writing A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and

examining the relation P^{−1}ABA^{−1}BP = I with the condition that det(A) = −1 we see there are no such matrices A ∈ GL(2, ℝ) that satisfy these properties. Thus this case gives no bundle-spaces.

We proceed by showing the bundle-spaces admit geometries as claimed. Obviously the product bundle Kb × T^{2} = (I, I; ( \begin{matrix}0 \\ 0 \end{matrix} )) is geometric of type E^{4}.

The bundle-space (I, I; ( \begin{matrix}1 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h), \\ αβα^{−1}β = l, & β(l, h)β^{−1} = (l, h). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨e_{1}, 2e_{2} −e_{3}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 \frac{1}{2} \frac{1}{2} 0 ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, l, h⟩ of Π_{1}(M) has index 2 and maps faithfully into ℝ^{4}

under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 1 \end{matrix} ) , ( 0 \frac{1}{2} \frac{1}{2} 0 ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus

this representation is both faithful and discrete with cocompact image so that (I, I; \binom{1}{0}) admits an E^{4} geometry.

The family of bundle-spaces (( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}(1) \\ (1) \end{matrix} ))_{λ≠0} admit geometries of type

Nil^{3} × E.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, l^{λ}h), \\ αβα^{−1}β = l^{(1)}h^{(1)}, & β(l, h)β^{−1} = (l, h). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(l)(x, y, z, w) = (x, y, z + 1, w),
i.e Φ(l) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(h)(x, y, z, w) = (x + h_{0}, y, z + h_{0}y, w)
i.e Φ(h) = (( \begin{matrix}1 & h_{0} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(β)(x, y, z, w) = (x + b_{0}, y, z + b_{0}y + b_{1}, w + 1)
i.e Φ(β) = (( \begin{matrix}1 & b_{0} & b_{1} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (1, 1)).
Φ(α)(x, y, z, w) = (x, y + 1, z, −w)
i.e Φ(α) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯));

where h_{0} is non-zero.

The relations [l, h] = 1, β(l, h)β^{−1} = (l, h) and αlα^{−1} = l are automatically satisfied. The relation αhα^{−1} = l^{λ}h determines that h_{0} = −λ; and the remaining relation αβα^{−1}β = l^{a}h^{b} implies that b_{0} = −bλ/2 and b_{1} = \frac{1}{4}(2a − λb).

When b = 0 to see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(l), Φ(β)⟩ = ⟨(z, w) ↦ (z + 1, w), (z, w) ↦ z + a/2, w + 1)⟩ ≅ ℤ^{2}. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient T^{2}. Similarly

\overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)↦^{[α]}(x, y + 1), (x, y)↦^{[h]}(x − λ, y)⟩ ≅ ℤ^{2} acts on ℝ^{2}⟨x, y⟩ with

quotient T^{2} so we have the desired representation when b = 0.

When b = 1 the group Φ(K) = ⟨(z, w)↦^{[l]}(z + 1, w), (z, w) ⟼^{[β^{2}h^{−1}]}(z, w + 1)⟩ ≅

ℤ^{2}; and acts on ℝ^{2}⟨z, w⟩ with quotient T^{2}. And the group \overline{Φ}(Π_{1}(M)/K) =

⟨(x, y)↦^{[α]}(x, y + 1), (x, y) ⟼^{[β^{2}=h]}(x − λ, y)⟩ is similarly isomorphic to ℤ^{2} and acts on ℝ^{2}⟨x, y⟩ with quotient T^{2}. Thus the family (( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , I; ( \begin{matrix}(1) \\ (1) \end{matrix} ))_{λ≠0} admit geometries

of type Nil^{3} × E. In fact we have shown that this family of bundle-spaces also fibre over T^{2} with fibre T^{2}.

The bundle-space (I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 0 \end{matrix} )) admits a geometry of type E^{4}.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h), \\ αβα^{−1}β = 1, & β(l, h)β^{−1} = (l^{−1}, h). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 0 \end{matrix} )⟩ ≅ ℤ^{4}. Thus this representation is both faithful and discrete with cocompact image so that (I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry. The bundle-space (I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}1 \\ 0 \end{matrix} )) admits a geometry of type E^{4}.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h), \\ αβα^{−1}β = l, & β(l, h)β^{−1} = (l^{−1}, h). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ −1 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨2e_{1} −e_{3}, e_{2}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \frac{1}{2} 0 \frac{1}{2} 0 ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}1 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 0 \end{matrix} )⟩ ≅ ℤ^{4}. Thus this representation is both faithful and discrete with cocompact image so that (I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}1 \\ 0 \end{matrix} )) admits an E^{4} geometry.

The bundle-space (I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 1 \end{matrix} )) admits a geometry of type E^{4}.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h), \\ αβα^{−1}β = h, & β(l, h)β^{−1} = (l^{−1}, h). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨e_{1}, 2e_{2} −e_{4}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 \frac{1}{2} 0 \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4}

under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ 1 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 0 \end{matrix} )⟩ ≅ ℤ^{4}. Thus

this representation is both faithful and discrete with cocompact image so that (I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 1 \end{matrix} )) admits an E^{4} geometry.

The bundle-space (I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}1 \\ 1 \end{matrix} )) admits a geometry of type E^{4}.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h), \\ αβα^{−1}β = lh, & β(l, h)β^{−1} = (l^{−1}, h). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ −1 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨2e_{1} − e_{3}, 2e_{2} − e_{4}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 \frac{1}{2} 0 \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \frac{1}{2} 0 \frac{1}{2} 0 ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

ON FIBRE-BUNDLES WITH BASE AND FIBRE EUCLIDEAN SURFACES 35

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}1 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ 1 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus this representation is both faithful and discrete with cocompact image so that (I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}1 \\ 1 \end{matrix} )) admits an E^{4} geometry. The bundle-space (I, ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 0 \end{matrix} )) admits a geometry of type E^{4}.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h), \\ αβα^{−1}β = 1, & β(l, h)β^{−1} = (l^{−1}, lh). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨e_{1}, e_{2}−2e_{3}, e_{3}, e_{3}+ 2e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( 0 ; 0 ; − \frac{1}{2} ; \frac{1}{2} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( 0 ; 0 ; − \frac{1}{2} ; \frac{1}{2} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (I, ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry. The bundle-space (I, ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 1 \end{matrix} )) admits a geometry of type E^{4}.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h), \\ αβα^{−1}β = h, & β(l, h)β^{−1} = (l^{−1}, lh). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & −1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨4e_{1} + e_{3}, 2e_{2} − e_{4}, e_{3}, −2e_{1} +e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \frac{1}{2} ; 0 ; − \frac{1}{2} ; 1 ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \frac{1}{4} ; \frac{1}{2} ; − \frac{1}{4} ; \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \frac{1}{4} ; 0 ; − \frac{1}{4} ; 0 ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \frac{1}{2} 1 0 1 ) , ( \frac{1}{2} ; 0 ; − \frac{1}{2} ; 0 ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \frac{1}{2} ; 0 ; − \frac{1}{2} ; 1 )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (I, ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 1 \end{matrix} )) admits an E^{4} geometry.

The bundle-space (−I, I; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h^{−1}), \\ αβα^{−1}β = 1, & β(l, h)β^{−1} = (l, h). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, l, h⟩ of Π_{1}(M) has index 2 and maps faithfully into ℝ^{4}

under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus

this representation is both faithful and discrete with cocompact image so that (I, −I; \binom{0}{0}) admits an E^{4} geometry.

The bundle-space (−I, I; ( \begin{matrix}1 \\ 0 \end{matrix} )) admits a Nil^{3} × E geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h^{−1}), \\ αβα^{−1}β = l, & β(l, h)β^{−1} = (l, h). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(l)(x, y, z, w) = (x, y, z − 1, w),
i.e Φ(l) = (( \begin{matrix}1 & 0 & −1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(h)(x, y, z, w) = (x, y, z, w + 1)
i.e Φ(h) = (I, (1, 1)).
Φ(β)(x, y, z, w) = (x, y + 1, z, w)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(α)(x, y, z, w) = (x + 1, −y, −z − y, −w)
i.e Φ(α) = (( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯))ρ.

Here ρ is the automorphism (x, y, z, w)↦^{ρ}(x, −y, −z, w).

All the relations are automatically satisfied. To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(l), Φ(h)⟩ ≅ ℤ^{2}. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient T^{2}. Similarly \overline{Φ}(Π_{1}(M)/K) =

⟨(x, y)↦^{[α]}(x + 1, −y), (x, y)↦^{[β]}(x, y + 1)⟩ ≅ ℤ˜×ℤ acts on ℝ^{2}⟨x, y⟩ with quotient Kb

so we have the desired representation.

The bundle-space (−I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h^{−1}), \\ αβα^{−1}β = 1, & β(l, h)β^{−1} = (l^{−1}, h). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4}

under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus

this representation is both faithful and discrete with cocompact image so that (−I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

The bundle-space (−I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}1 \\ 0 \end{matrix} )) admits a geometry of type E^{4}.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h^{−1}), \\ αβα^{−1}β = l, & β(l, h)β^{−1} = (l^{−1}, h). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨e_{1}, 2e_{2} +e_{3}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 ; \frac{1}{2} ; − \frac{1}{2} ; 0 ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( 0 1 \frac{1}{2} 0 ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus this representation is both faithful and discrete with cocompact image so that (−I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}1 \\ 0 \end{matrix} )) admits an E^{4} geometry. The family of bundle-spaces (−I, ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}0 \\ n \end{matrix} ))_{n∈ℕ\\{0\}} admit geometries of type Nil^{3} × E.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h^{−1}), \\ αβα^{−1}β = h^{n}, & β(l, h)β^{−1} = (l^{−1}, lh). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(l)(x, y, z, w) = (x, y, z, w + 1)
i.e Φ(h) = (I, (1, 1)).
Φ(h)(x, y, z, w) = (x, y, z + 1, w + k),
i.e Φ(l) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (k, 1)).
Φ(β)(x, y, z, w) = (x, y + 1, z, −w)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯)).
Φ(α)(x, y, z, w) = (x + 1, −y, −z − y + a_{0}, −w + a_{1})
i.e Φ(α) = (( \begin{matrix}1 & 1 & a_{0} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (a_{1}, 𝔯))ρ.

Here ρ is the automorphism (x, y, z, w)↦^{ρ}(x, −y, −z, w).

The relations [l, h] = 1, α(l, h)α^{−1} = (l^{−1}, h^{−1}) and βlβ^{−1} = l^{−1} are automatically satisfied. The relation βhβ^{−1} = lh implies that k = −1/2; and the relation αβα^{−1}β = h^{n} gives the relations a_{0} = (n + 1)/2 and a_{1} = −n/4. To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(l), Φ(h)⟩ ≅ ℤ^{2}. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient T^{2}.

Similarly \overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)↦^{[α]}(x + 1, −y), (x, y)↦^{[β]}(x, y + 1)⟩ ≅ ℤ˜×ℤ acts on ℝ^{2}⟨x, y⟩ with quotient Kb so we have the desired representation.

The Lie group Nil^{4} is the semi-direct product of ℝ^{3} = \{( \begin{matrix}x \\ y \\ z \end{matrix} )\} and ℝ⟨t⟩ where

the action of t ∈ ℝ is left multiplication by the matrix ( \begin{matrix}1 & t & t^{2}/2 \\ 0 & 1 & t \\ 0 & 0 & 1 \end{matrix} ). The identity

component Isom^{0}(Nil^{4}) of the isometry group is again Nil^{4}. The quotient group Isom(Nil^{4})/Isom^{0}(Nil^{4}) is isomorphic to ℤ/2ℤ ⊕ ℤ/2ℤ generated by the automorphisms:

(x, y, z, t) ↦ (−x, −y, −z, t), (x, y, z, t) ↦ (x, −y, z, −t).

The commutator subgroup of Nil^{4} = Isom^{0}(Nil^{4}) is ℝ^{2}⟨x, y⟩ so abelianization gives Nil^{4} the structure of a plane-bundle over ℝ^{2}⟨z, t⟩:

\xymatrix{1 \ar[r] & ℝ^{2}⟨x, y⟩ \ar[r] & Nil_{4} \ar[r] & ℝ^{2}⟨z, t⟩ \ar[r] & 1.}

Passing to isometries we see that the above automorphisms preserve the bundle structure so we have an exact sequence:

\xymatrix{1 \ar[r] & Isom(ℝ^{2}) \ar[r] & Isom(Nil^{4}) \ar[r] & Isom(ℝ^{2}) \ar[r] & 1.}

Lemma. Let [figure: ℘: Isom(Nil^{4}); Isom(E^{2})] be the induced map of the ℝ^{2}-bundle

projection. A representation Φ: Π_{1}(M) → I som(Nil^{4}) is faithful and discrete with cocompact image precisely when the following two properties are satisfied:

(1) The subgroup K ⊴ Π_{1}(M) whose image Φ(K) = Φ(Π_{1}(M))∩ker(℘) embeds faithfully, discretely and cocompactly in I som(E^{2}).

(2) The quotient group Π_{1}(M)/K embeds faithfully and discretely in I som(E^{2}) with cocompact image under the induced map from Φ.

proof: The proof is entirely analogous to that of the Nil^{3} × E or Sol^{3} × E case. □

The family of bundle-spaces (( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ) , I; ( \begin{matrix}m \\ n \end{matrix} ))_{λ,n≠0} admit geometries of

type Nil^{4}.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, l^{λ}h^{−1}), \\ αβα^{−1}β = l^{m}h^{n}, & β(l, h)β^{−1} = (l, h). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → Isom(Nil^{4}) as follows:

Φ(l)(x, y, z, t) = (x + l_{0}, y, z, t),
i.e Φ(l) = (( \begin{matrix}l_{0} \\ 0 \\ 0 \end{matrix} ) , 0).
Φ(h)(x, y, z, t) = (x, y + h_{0}, z, t)
i.e Φ(h) = (( \begin{matrix}0 \\ h_{0} \\ 0 \end{matrix} ) , 0).
Φ(β)(x, y, z, t) = (x, y + b_{1}, z + 1, t)
i.e Φ(β) = (( \begin{matrix}0 \\ b_{1} \\ 1 \end{matrix} ) , 0).
Φ(α)(x, y, z, t) = (−x − y − z/2, −y − z, −z, t + 1)
i.e Φ(α) = (( \begin{matrix}0 \\ 0 \\ 0 \end{matrix} ) , 1)ρ.

where h_{0}, l_{0} are non-zero and ρ is the automorphism (x, y, z, t)↦^{ρ}(−x, −y, −z, t).

The relations [l, h] = 1, αlα^{−1} = l^{−1} and β(l, h)β^{−1} = (l, h) are automatically satisfied. The relation αhα^{−1} = l^{λ}h^{−1} implies that l_{0} = −h_{0}/λ; and the relation αβα^{−1}β = l^{m}h^{n} gives the relations h_{0} = −1/n and b_{1} = −ml_{0} − 1/2. Therefore l_{0} = 1/nλ and b_{1} = −m/nλ − 1/2.

Here the group Φ(K) = Stab_{Φ(Π_{1}(M))}⟨z, t⟩ is ⟨Φ(l), Φ(h)⟩ = ⟨(x, y) ↦ (x + 1/nλ, y), (x, y) ↦ (x, y − 1/n)⟩ ≅ ℤ^{2} and acts on ℝ^{2}⟨x, y⟩ with quotient T^{2}. The

group \overline{Φ}(Π_{1}(M)/K) = ⟨(z, t)↦^{[α]}(z+1, t), (z, t)↦^{[β]}(−z, t+1) is isomorphic to ℤ˜×ℤ and acts on ℝ^{2}⟨z, t⟩ with quotient the Klein bottle. Thus this representation is faithful and discrete with cocompact image so that the family (( \begin{matrix}−1 & λ \\ 0 & −1 \end{matrix} ) , I; ( \begin{matrix}m \\ n \end{matrix} ))_{λ,n≠0}

admit geometries of type Nil^{4}.

The bundle-space (I, −I; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h), \\ αβα^{−1}β = 1, & β(l, h)β^{−1} = (l^{−1}, h^{−1}). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as follows:

\begin{matrix}l & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , I), \\ h & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, l, h⟩ of Π_{1}(M) is of finite index and maps faithfully into ℝ^{4}

under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus

this representation is both faithful and discrete with cocompact image so that (I, −I; \binom{0}{0}) admits an E^{4} geometry.

The bundle-space (I, −I; ( \begin{matrix}1 \\ 0 \end{matrix} )) admits a Nil^{3} × E geometry.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h), \\ αβα^{−1}β = l, & β(l, h)β^{−1} = (l^{−1}, h^{−1}). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(l)(x, y, z, w) = (x, y, z + 1, w),
i.e Φ(l) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(h)(x, y, z, w) = (x, y, z, w + 1)
i.e Φ(h) = (I, (1, 1)).
Φ(β)(x, y, z, w) = (x, −y + 1, −z, −w)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯))ρ.
Φ(α)(x, y, z, w) = (x + 1, y, z + y, w)
i.e Φ(α) = (( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).

Here ρ is the automorphism (x, y, z, w)↦^{ρ}(x, −y, −z, w).

All the relations are automatically satisfied. To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(l), Φ(h)⟩ ≅ ℤ^{2}. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient T^{2}. Similarly \overline{Φ}(Π_{1}(M)/K) =

⟨(x, y)↦^{[α]}(x + 1, y), (x, y)↦^{[β]}(x, −y + 1)⟩ ≅ ℤ˜×ℤ acts on ℝ^{2}⟨x, y⟩ with quotient Kb

so we have the desired representation.

The family of bundle-spaces (( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}(1) \\ (1) \end{matrix} ))_{λ≠0} admit geometries of

type Nil^{3} × E.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, l^{λ}h), \\ αβα^{−1}β = l^{(1)}h^{(1)}, & β(l, h)β^{−1} = (l^{−1}, h^{−1}). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(l)(x, y, z, w) = (x, y, z + 1, w),
i.e Φ(l) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(h)(x, y, z, w) = (x + h_{0}, y, z + h_{0}y, w)
i.e Φ(h) = (( \begin{matrix}1 & h_{0} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(β)(x, y, z, w) = (−x + b_{0}, y, −z + b_{0}y, w + 1)
i.e Φ(β) = (( \begin{matrix}1 & b_{0} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (1, 1))ρ.
Φ(α)(x, y, z, w) = (x + a_{0}, y + 1, z + a_{0}y, −w)
i.e Φ(α) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯));

where h_{0} is non-zero and ρ is the automorphism (x, y, z, w)↦^{ρ}(−x, y, −z, w).

The relations [l, h] = 1, β(l, h)β^{−1} = (l^{−1}, h^{−1}) and αlα^{−1} = l are automatically satisfied. The relation αhα^{−1} = l^{λ}h determines that h_{0} = −λ; and the remaining relation αβα^{−1}β = l^{a}h^{b} implies that a_{0} = −bλ/2, and b_{0} = bλ/2 − a.

To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(l), Φ(β^{2})⟩ = ⟨(z, w) ↦ (z + 1, w), (z, w) ↦ z, w + 2)⟩ ≅ ℤ^{2}. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient T^{2}. Similarly the group \overline{Φ}(Π_{1}(M)/K) =

⟨(x, y)↦^{[α]}(x − bλ/2, y + 1), (x, y)↦^{[β]}(−x + bλ/2 − a, y), (x, y)↦^{[h]}(x − λ, y)⟩ is isomor-

phic to the amalgamated free product (ℤ ⊕ ℤ/2ℤ) ∗_{ℤ} (ℤ ⊕ ℤ/2ℤ) when b = 0, and to ℤ ∗_{ℤ} (ℤ ⊕ ℤ/2ℤ) when b = 1; and acts on ℝ^{2}⟨x, y⟩ with quotient the silvered annulus SAn or the silvered Möbius band SMb respectively. Thus we have the desired representation and the family (( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) , −I; ( \begin{matrix}(1) \\ (1) \end{matrix} ))_{λ≠0} admit geometries of

type Nil^{3} × E. In fact we have shown that this family of bundle-spaces is not only Seifert fibered over Kb with fibre T^{2} but is also Seifert fibred over either SAn or SMb with general fibre T^{2}.

The family of bundle-spaces (A, ±I; ( \begin{matrix}a \\ b \end{matrix} ), det(A) = 1, |trace(A)| > 2) admit geometries of type Sol^{3} × E.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l, h)A, \\ αβα^{−1}β = l^{a}h^{b}, & β(l, h)β^{−1} = (l^{±1}, h^{±1}). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Sol^{3} × E) as follows: We have that P AP^{−1} = ({e_{0}^{t} \atop 0} {0 \atop e^{−t_{0}}} ) for some P = ({p \atop r} {q \atop s}) ∈ GL(2, ℝ). Hence we define a

representation Φ: Π_{1}(M) → I som(Sol^{3} × E) as follows:

Φ(l)(x, y, t, u) = (x + p, y + r, t, u)
ie Φ(l) = ((\binom{p}{q}, 0), (0, 1)).
Φ(h)(x, y, t, u) = (x + q, y + s, t, u)
ie Φ(h) = ((\binom{r}{s}, 0), (0, 1)).
Φ(β)(x, y, t, u) = (ϵx + x_{1}, ϵy + y_{1}, t, u + 1)
ie Φ(β) = ((\binom{x_{1}}{y_{1}}, 0), (1, 1))λ.
Φ(α)(x, y, t, u) = (e^{t_{0}}x + x_{0}, e^{−t_{0}}y + y_{0}, t + t_{0}, −u)
ie Φ(α) = ((\binom{x_{0}}{y_{0}}, t_{0}), (0, 𝔯)) t_{0} ≠ 0.

where ϵ = ±1 respective as to whether B = ±I and hence (x, y, t, u)⟼^{λ}(ϵx, ϵy, t, u).

The relations [l, h] = 1, α(l, h)α^{−1} = (l, h)A and β(l, h)β^{−1} = (l^{±1}, h^{±1}) are automatically satisfied. The relation [α, β] = l^{a}h^{b} gives the relation:

\binom{x_{1}}{y_{1}} = (I + ϵP AP^{−1})^{−1} ((ϵ − 1)\binom{x_{0}}{y_{0}} + P \binom{a}{b}) .

Here K = ⟨Φ(l), Φ(h)⟩. Since P ∈ GL(2, ℝ) the vectors ( \begin{matrix}p \\ r \end{matrix} ) and ( \begin{matrix}q \\ s \end{matrix} ) are linearly independent and hence K acts on ℝ^{2}⟨x, y⟩ with quotient T^{2}.

The family of bundle-spaces (( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , B; ( \begin{matrix}a \\ b \end{matrix} ), det(B) = 1, |trace(B)| > 2) admit geometries of type Sol^{3} × E.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (h, l^{−1}, \\ αβα^{−1}β = l^{a}h^{b}, & β(l, h)β^{−1} = (l, h)B. \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Sol^{3} × E) as follows: We have that

P BP^{−1} = ({e_{1}^{t} \atop 0} {0 \atop e^{−t_{1}}} ) for some P = ({p \atop r} {q \atop s}) ∈ GL(2, ℝ) that commutes with ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ).

Hence we define a representation Φ: Π_{1}(M) → I som(Sol^{3} × E) as follows:

Φ(l)(x, y, t, u) = (x + p, y + r, t, u)
ie Φ(l) = ((\binom{p}{q}, 0), (0, 1)).
Φ(h)(x, y, t, u) = (x + q, y + s, t, u)
ie Φ(h) = ((\binom{r}{s}, 0), (0, 1)).
Φ(β)(x, y, t, u) = (e^{t_{1}}x + x_{1}, e^{−t_{1}}y + y_{1}, t + t_{0}, u)
ie Φ(β) = ((\binom{x_{1}}{y_{1}}, t_{1}), (0, 1)).
Φ(α)(x, y, t, u) = (y + x_{0}, −x + y_{0}, −t, u + 1)
ie Φ(α) = ((\binom{x_{0}}{y_{0}}, t_{0}), (1, 1))ρ t_{0} ≠ 0.

Here ρ is the automorphism (x, y, t, u)⟼^{ρ}(y, −x, −t, u).

The relations [l, h] = 1, α(l, h)α^{−1} = (l, h)A and β(l, h)β^{−1} = (l^{±1}, h^{±1}) are automatically satisfied. The relation [α, β] = l^{a}h^{b} gives the relation:

\binom{x_{0}}{y_{0}} = P BP^{−1} ((P BP^{−1} − ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )) ( \begin{matrix}x_{1} \\ y_{1} \end{matrix} ) − P \binom{a}{b}) .

Here K = ⟨Φ(l), Φ(h)⟩. Since P ∈ GL(2, ℝ) the vectors ( \begin{matrix}p \\ r \end{matrix} ) and ( \begin{matrix}q \\ s \end{matrix} ) are linearly independent and hence K acts on ℝ^{2}⟨x, y⟩ with quotient T^{2}.

The families of bundle-spaces (−I, ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ); ( \begin{matrix}(1) \\ n \end{matrix} ))_{n∈ℕ\\{0\}} admit geometries of type Nil^{3} × E.

Π_{1}(M) = ⟨α, β, l, h | \begin{matrix}[l, h] = 1, & α(l, h)α^{−1} = (l^{−1}, h^{−1}), \\ αβα^{−1}β = l^{(}1)h^{n}, & β(l, h)β^{−1} = (l^{−1}, h). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(l)(x, y, z, w) = (x, y, z, w + 1)
i.e Φ(h) = (I, (1, 1)).
Φ(h)(x, y, z, w) = (x, y, z + 1, w),
i.e Φ(l) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(β)(x, y, z, w) = (x, y + k, z, −w)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & k \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯)).
Φ(α)(x, y, z, w) = (x + 1, −y, −z − y, −w + a)
i.e Φ(α) = (( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (a, 𝔯))ρ.

Here ρ is the automorphism (x, y, z, w)↦^{ρ}(x, −y, −z, w).

The relations [l, h] = 1, α(l, h)α^{−1} = (l^{−1}, h^{−1}) and β(l, h)β^{−1} = (l^{−1}, h) are automatically satisfied. The relation αβα^{−1}β = l^{(1)}h^{n} implies that a = (1)/2 and k = −n. To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(l), Φ(h)⟩ ≅ ℤ^{2}. Clearly K acts on ℝ^{2}⟨z, w⟩ with

quotient T^{2}. Similarly \overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)⟼^{[α]}(x+1, −y), (x, y)⟼^{[β]}(x, y −n)⟩ ≅ ℤ˜×ℤ acts on ℝ^{2}⟨x, y⟩ with quotient Kb so we have the desired representation.

The Classification of Total Spaces of Kb-Bundles over T^{2} or Kb and their Geometries

Let M be the total space of a Kb-bundle over the surface S where S is either T^{2} or Kb. If S = T^{2} then the fundamental group of M is

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = ϑ(α)(γ, δ), \\ [α, β] = γ^{m}δ^{n}, & β(γ, δ)β^{−1} = ϑ(β)(γ, δ). \end{matrix} ⟩ .

and extension equivalences are in bijection with:

H^{2}(ℤ^{2}; ℤ) ≅ ℤ/⟨κ(α) − 1, κ(β) − 1⟩

If S = Kb then the fundamental group of M is

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = ϑ(α)(γ, δ), \\ αβα^{−1}β = γ^{m}δ^{n}, & β(γ, δ)β^{−1} = ϑ(β)(γ, δ). \end{matrix} ⟩ .

and extension equivalences are in bijection with:

H^{2}(ℤ^{2}; ℤ) ≅ ℤ/⟨−κ(α) − 1, κ(β) − 1⟩

where κ is the action on the center of ℤ) ≅ ℤ induced by ϑ.

Here the mapping class group of the fibre Kb is the group Out(Π_{1}(Kb)) ≅ ℤ/2ℤ ⊕ ℤ/2ℤ which we take to be the four automorphisms:

\begin{matrix}(γ, δ) & ⟼^{e}(γ, δ), \\ (γ, δ) & ⟼^{φ_{1}}(γδ, δ), \end{matrix}  \begin{matrix}(γ, δ) & ⟼^{φ_{2}}(γ^{−1}, δ), \\ (γ, δ) & ⟼^{φ_{3}}(γ^{−1}δ, δ). \end{matrix}

These representatives genuinely commute as automorphisms of Π_{1}(Kb), however whilst φ_{2}^{2} is the identity automorphism φ_{1} and φ_{3} square to give non-trivial inner automorphisms. Note that e and φ_{1} act identically on Z(Π_{1}(F)) = ⟨γ^{2}⟩ whilst the maps φ_{2} and φ_{3} act on the center by inversion.

When the base-space is T^{2}, the action given by ϑ([α, β]) is the identity inner automorphism, and hence is well-defined up to multiplication by central elements of Π_{1}(F). Therefore the Euler number must be central so that ( \begin{matrix}m \\ n \end{matrix} ) ∈ ⟨({2 \atop 0})⟩ ≅ Z(Π_{1}(Kb)).

When the base space is Kb, the action given by ϑ(αβα^{−1}β) = ϑ(β^{2}) is the identity inner automorphism when ϑ(β) = e or φ_{2}; and in this case, as above the Euler number must represent a central element of Π_{1}(F). However when ϑ(β) = φ_{1} or φ_{3}, the action ϑ(αβα^{−1}β) is by conjugation by δ^{−1}; and therefore the Euler number will lie in the coset of Z(Π_{1}(F)) represented by δ^{−1}. That is the Euler number must be of the form ( \begin{matrix}m \\ n \end{matrix} ) ∈ \{({2k \atop 1} )\} where k is free in ℤ.

Note that the orientation character w_{1}(M): Π_{1}(M) → Out(Π_{1}(F)) always restricts nontrivially to ⟨γ⟩ so that these bundle-spaces are all non-orientable. Such Kb-bundle-spaces are classified by the following two theorems.

Theorem. Let M be the total space of a Kb-bundle over T^{2} with first Betti number β_{1}(M). Then M is bundle diffeomorphic to precisely one of the geometric 4-manifolds listed below:

Total Space  Geometric Type

β_{1}(M) = 3  \begin{matrix}(e, e; ({0 \atop 0})) & E^{4} \\ (φ_{1}, e; ({0 \atop 0})) & E^{4} \end{matrix}

β_{1}(M) = 2  \begin{matrix}(e, e; ( \begin{matrix}2n \\ 0 \end{matrix} ))_{n∈ℕ\\{0\}} & Nil^{3} × E \\ (φ_{1}, e; ( \begin{matrix}2n \\ 0 \end{matrix} ))_{n∈ℕ\\{0\}} & Nil^{3} × E \\ (φ_{2}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) & E^{4} \\ (φ_{3}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) & E^{4} \\ (φ_{1}, φ_{2}; ( \begin{matrix}(2) \\ 0 \end{matrix} )) & E^{4} \end{matrix}

Table 4.

Above and henceforth the symbol (2) is used to denote a fixed integer equal to 0 or 2.

Theorem. Let M be the total space of a Kb-bundle over Kb with first Betti number β_{1}(M). Then M is bundle diffeomorphic to precisely one of the geometric 4-manifolds listed below:

Total Space  Geometric Type

β_{1}(M) = 2  \begin{matrix}(e, e; ({(2) \atop 0})) & E^{4} \\ (φ_{1}, e; ({(2) \atop 0})) & E^{4} \\ (e, φ_{1}; ({(2) \atop −1})) & Nil^{3} × E \end{matrix}

β_{1}(M) = 1

(φ_{2}, e; ( \begin{matrix}0 \\ 0 \end{matrix} ))  E^{4}
(φ_{2}, e; ( \begin{matrix}2n \\ 0 \end{matrix} ))_{n∈ℕ\\{0\}}  Nil^{3} × E
(φ_{3}, e; ( \begin{matrix}0 \\ 0 \end{matrix} ))  E^{4}
(φ_{3}, e; ( \begin{matrix}2n \\ 0 \end{matrix} ))_{n∈ℕ\\{0\}}  Nil^{3} × E
(e, φ_{2}; ( \begin{matrix}(2) \\ 0 \end{matrix} ))  E^{4}
(e, φ_{3}; ( \begin{matrix}(2) \\ −1 \end{matrix} ))  E^{4}
(φ_{2}, φ_{1}; ( \begin{matrix}0 \\ −1 \end{matrix} ))  E^{4}
(φ_{2}, φ_{1}; ( \begin{matrix}2n \\ −1 \end{matrix} ))_{n∈ℕ\\{0\}}  Nil^{3} × E
(φ_{1}, φ_{2}; ( \begin{matrix}(2) \\ 0 \end{matrix} ))  E^{4}
(φ_{1}, φ_{3}; ( \begin{matrix}(2) \\ −1 \end{matrix} ))  E^{4}

Table 5.

proof.

Abelianizing the presentation for Π_{1}(M) and employing the function r: Out(Π_{1}(Kb)) → GL(2, ℤ) defined by:

e ↦ I, φ_{1} ↦ ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) , φ_{2} ↦ ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , φ_{3} ↦ ( \begin{matrix}−1 & 0 \\ 1 & 1 \end{matrix} ) ,

we see that

β_{1}(M) = \{ \begin{matrix}4 − rank(( \begin{matrix}0 & m \\ 2 & n \end{matrix} ) , rϑ(α) − I, rϑ(β) − I), & if S = T^{2}, \\ 3 − rank(( \begin{matrix}0 \\ 2 \end{matrix} ) , rϑ(α) − I, rϑ(β) − I), & if S = Kb. \end{matrix}

To obtain a listing of all such bundle-spaces our approach is to ‘build up’ the 4-manifold M as the bundle space of a 3-manifold bundle over the circle. This is done by firstly utilizing the S^{1}-bundle over S^{1} structure of the base space S so as to induce a Kb over the fibre of this bundle with total space B a 3-manifold. Then M fibres over S^{1} with fibre B. Known information about Π_{1}(B) shall allow us to classify Π_{1}(M) whilst preserving the Kb-bundle structure of M over S. This is roughly summarized in the diagram below:

[figure: Kb; B; S^{1}; Kb; M; S; S^{1}; S^{1}]

We proceed in greater detail below.

From the long exact homotopy sequence for a fibration we have:

[figure: 1; Π_{1}(Kb); Π_{1}(M); p; Π_{1}(S); 1; ⟨γ, δ⟩; ⟨α, β, γ, δ⟩; ⟨α, β⟩.]

Moreover there is a projection q: Π_{1}(S) ↠ ℤ⟨α⟩ so that we have an epimorphism qp: Π_{1}(M) ↠ ℤ whose kernel ker(qp) = ⟨β, γ, δ⟩ is the fundamental group of the total space of a Kb-bundle over S^{1} with homotopy sequence:

[figure: 1; Π_{1}(Kb); ker(qp); \overline{p}; Π_{1}(S^{1}); 1; ⟨γ, δ⟩; ⟨β, γ, δ⟩; ⟨β⟩.]

The image of β under \overline{p} is a generator for ℤ and hence determines a right inverse s to \overline{p} so that this sequence splits and ker(qp) ≅ Π_{1}(Kb) ⋊_{θ} ℤ⟨β⟩ where θ = im(ϑ|_{⟨β⟩}) is in the mapping class group of the fibre Out(Π_{1}(Kb)) and is determined by the action of conjugation of β on γ and δ. That is ker(pq) is one of the four non-orientable flat 3-manifold groups: B_{1} −B_{4}, where B_{i+1} ≅ Π_{1}(Kb)⋊_{φ_{i}} ℤ and φ_{0} = e; coresponding to the elements of Out(Π_{1}(Kb)) ≅ ℤ/2ℤ ⊕ ℤ/2ℤ; (see [Hi1] (and also [Wo])).

Hence M fibres over S^{1} with fibre some non-orientable flat 3-manifold represented by the split exact sequence:

[figure: 1; ker(qp); Π_{1}(M); qp; Π_{1}(S^{1}); 1; ⟨β, γ, δ⟩; ⟨α, β, γ, δ⟩; ⟨α⟩.]

Therefore Π_{1}(M) ≅ ker(qp) ⋊_{ψ} ℤ where ψ ∈ Out(ker(qp)) is determined by the action of conjugation of α on β, γ and δ. Note that ψ|_{⟨γ,δ⟩} = im(ϑ|_{⟨α⟩}).

Kb-bundles over T^{2}.

We proceed in three cases according to the image of ϑ: ℤ^{2} → (ℤ/2ℤ)^{2}.

The action of Π_{1}(T^{2}) on Π_{1}(Kb) is trivial.

Here we have an infinite family of bundle equivalences with respect to this action since H^{2}(ℤ^{2}; ℤ) ≅ ℤ.

In this case ϑ acts by inner automorphisms. In particular θ is trivial so that ker(qp) is the direct product Π_{1}(Kb)×ℤ = B_{1}. Therefore Π_{1}(M) = B_{1} ⋊_{ψ} ℤ where ψ ∈ Out(B_{1}) and restricts trivially to ⟨γ, δ⟩ < B_{1}. Now Out(B_{1}) ≅ Γ_{2} × ℤ/2ℤ where Γ_{2} is the subgroup of GL(2, ℤ) consisting of matrices congruent mod(2) to upper triangular matrices. Such automorphisms are of the form:

(β, γ, δ) ⟼ (β^{a}γ^{b}, β^{c}γ^{d}δ^{(1)}, δ);

where ( \begin{matrix}a & c \\ b & d \end{matrix} ) ∈ Γ_{2} and (1) ∈ ℤ/2ℤ. Since ψ acts trivially on γ we have that c = 0, d = 1 and (1) = 0. Furthermore β and γ commute in B_{1} so that the action of ψ on β corresponds to the relation αβα^{−1} = γ^{b}β^{a}. Thus a = 1 and b is free in 2ℤ. And we have an infinite family of representations with trivial monodromy over the base and Euler number ( \begin{matrix}n \\ 0 \end{matrix} ) where n is even. However changing the base of the fibre by inverting γ, we may assume that n is positive. That is we obtain the family of characteristic triples:

(e, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℕ}.

Each member of this family indeed represents a distinct bundle-space as each group abelianizes as: H_{1}(M) ≅ ℤ^{2} ⊕ ℤ/2ℤ ⊕ ℤ/2nℤ.

The action of Π_{1}(T^{2}) on Π_{1}(Kb) is via a non-trival proper subgroup of Out(Π_{1}(Kb)).

The only non-trivial proper subgroup of ℤ/2ℤ ⊕ ℤ/2ℤ up to group isomorphism is ℤ/2ℤ. Therefore by performing a basis change on Π_{1}(T^{2}) we may assume that β ∈ ker(ϑ) so that the image of α under ϑ is ℤ/2ℤ. Then as before θ is trivial so that ker(qp) = B_{1} and Π_{1}(M) ≅ B_{1} ⋊_{ψ} ℤ where the restriction of ψ ∈ Out(B_{1}) to ⟨γ, δ⟩ is non-trivial. Thus a = 1 and the Euler number is ( \begin{matrix}n \\ 0 \end{matrix} )_{n∈2ℤ}. Also since ψ|_{⟨α⟩} ∈ Out(Π_{1}(Kb)) we must have c = 0; and all three non-trivial outer automorphisms φ_{1}, φ_{2}, φ_{3} of Π_{1}(Kb) may be realized as im(ϑ|_{⟨α⟩}) by choosing d and (1) appropriately. As with the infinite family of triples above we may assume that n ≥ 0. That is we obtain the three infinite families:

(φ_{1}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℕ}, (φ_{2}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℕ}, (φ_{3}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℕ}. These representations admit

further refinement as equivalence classes of the corresponding group extensions are counted by elements of H^{2}(ℤ^{2}; ℤ); being isomorphic to ℤ, ℤ/2ℤ or ℤ/2ℤ respectively.

In the case where im(ϑ|_{⟨α⟩}) = φ_{1} the infinite family of characteristic triples represents an infinite family of distinct bundle-spaces. This is easily seen by calculating the first homology group of the bundle-space to be: H_{1}(M) ≅ ℤ^{2} ⊕ ℤ/2nℤ.

However in the case where im(ϑ|_{⟨α⟩}) = φ_{2} we may apply a change of crosssection of the form (α, β) ↦ (α, βγ^{2k}) for some appropriate k ∈ ℤ reducing the Euler number to one of the two possibilities: ( \begin{matrix}(2) \\ 0 \end{matrix} ). These two characteristic triples represent distinct bundle-spaces as the corresponding fundamental groups have distinct centers being ⟨β⟩ ≅ ℤ or trivial respective of whether (2) = 0 or 2.

Similarly in the case where im(ϑ|_{⟨α⟩}) = φ_{3} isomorphisms as above permit us to

transform the Euler number from ( \begin{matrix}2n′ \\ 0 \end{matrix} ) to ( \begin{matrix}(2) \\ 0 \end{matrix} ). The resulting two groups are distinguished by the center; being ⟨β⟩ ≅ ℤ or trivial respective of whether (2) = 0 or 2. Thus this pair of triples indeed represent distinct bundle-spaces. That is we obtain one infinite family and two pairs of characteristic triples:

(φ_{1}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℤ}, (φ_{2}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )), (φ_{3}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )).

The action of Π_{1}(T^{2}) on Π_{1}(Kb) is via the whole group Out(Π_{1}(Kb)).

Up to a change of basis of ℤ^{2} there is only one group epimorphism ℤ^{2}↠^{ϑ}(ℤ/2ℤ)^{2}. Hence applying a basis change to Π_{1}(T^{2}) we may assume that the image of ϑ|_{⟨α⟩} is ⟨φ_{1}⟩ and hence the image of ϑ|_{⟨β⟩} is ⟨φ_{2}⟩. Therefore ker(qp) ≅ Π_{1}(Kb) ⋊_{φ_{2}} ℤ = B_{3} and Π_{(}M) ≅ B_{3} ⋊_{ψ} ℤ where ψ ∈ Out(B_{3}) coincides with φ_{1} when restricted to ⟨γ, δ⟩. Here Out(B_{3}) ≅ (ℤ/2ℤ)^{3} and is generated by the automorphisms:

(β, γ, δ)⟼^{μ_{1}}(β^{−1}, γ, δ), (β, γ, δ)⟼^{μ_{2}}(β, γδ, δ), (β, γ, δ)⟼^{μ_{3}}(βγ^{2}, γ, δ).

Note. There is misprint in [H1] where an automorphism corresponding to μ_{2} appears as an inner automorpism of B_{3}.

Hence we see the only possiblities for ψ are μ_{2} or μ_{2}μ_{3}. Thus noting that βγ^{2} = γ^{−2}β we obtain the two possible Euler numbers: ( \begin{matrix}(−2) \\ 0 \end{matrix} ). However we may

change the sign of the Euler number by applying a transformation (γ, δ) ↦ (γ^{−1}, δ) to the fibre. Thus we obtain the two representations:

(φ_{1}, φ_{2}; ( \begin{matrix}0 \\ 0 \end{matrix} )), (φ_{1}, φ_{2}; ( \begin{matrix}2 \\ 0 \end{matrix} )).

These extensions come from distinct equivalence classes as H^{2}(ℤ^{2}; ℤ) ≅ ℤ/2ℤ. Infact these two triples represent distinct bundle-spaces since the center of Π_{1}(M) is alternately ⟨β^{2}⟩ ≅ ℤ or trivial depending on whether the Euler number is respectively trivial or not.

Thus Table 4 contains a complete non-redundant listing of all Kb-bundles over T^{2}. We now show these bundle-spaces admit geometries as claimed.

The bundle-space (φ_{1}, e; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γδ, δ), \\ [α, β] = 1, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , I). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨2e_{1} +e_{4}, e_{2}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \frac{1}{2} ; 0 ; 0 ; − \frac{1}{2} ) , I). \end{matrix}

The subgroup A⟨α, β, γ^{2}, δ⟩ of Π_{1}(M) has index 2 and maps faithfully into ℝ^{4}

under Φ with discrete, cocompact image ⟨( \frac{1}{2} ; 0 ; 0 ; − \frac{1}{2} ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus

this representation is both faithful and discrete with cocompact image so that (φ_{1}, e; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

The family of bundle-spaces (e, e; ( \begin{matrix}2n \\ 0 \end{matrix} ))_{n∈ℕ\\{0\}} admit geometries of type

Nil^{3} × E.

Remark. It is clear that when n = 0 the bundle-space M ≃ T^{2} × Kb admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ, δ), \\ [α, β] = γ^{2n}, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(γ)(x, y, z, w) = (x, y, z + 1, −w),
i.e Φ(γ) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯)).
Φ(δ)(x, y, z, w) = (x, y, z, w + 1)
i.e Φ(δ) = (I, (1, 1)).
Φ(β)(x, y, z, w) = (x, y + b, z, w)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(α)(x, y, z, w) = (x + 1, y, z + y, w)
i.e Φ(α) = (( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1));

where b is non-zero.

Then the relation γδγ^{−1}δ = 1 and the commutativity of α and β with γ and δ are preserved by Φ. The remaining relation [α, β] = γ^{2n} implies that b = 2n.

To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(γ), Φ(δ)⟩ = ⟨(z, w) ↦ (z + 1, −w), (z, w) ↦ z, w + 1)⟩ ≅ ℤ˜×ℤ. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient Kb. Similarly

\overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)↦^{[α]}(x + 1, y), (x, y)↦^{[β]}(x, y + 2n)⟩ ≅ ℤ^{2} acts on ℝ^{2}⟨x, y⟩ with

quotient T^{2} so we have the desired representation.

The family of bundle-spaces (φ_{1}, e; ( \begin{matrix}2n \\ 0 \end{matrix} ))_{n∈ℕ\\{0\}} admit geometries of type

Nil^{3} × E.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γδ, δ), \\ [α, β] = γ^{2n}, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(γ)(x, y, z, w) = (x, y, z + 1, −w),
i.e Φ(γ) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯)).
Φ(δ)(x, y, z, w) = (x, y, z, w + 1)
i.e Φ(δ) = (I, (1, 1)).
Φ(β)(x, y, z, w) = (x, y + b, z, w)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(α)(x, y, z, w) = (x + 1, y, z + y, w + a)
i.e Φ(α) = (( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (a, 1));

where b is non-zero.

Then the relations γδγ^{−1}δ = 1, αδα^{−1} = δ and β(γ, δ)β^{−1} = (γ, δ) are preserved by Φ. The relation αγα^{−1} = γδ determines that a = 1/2; and the remaining relation [α, β] = γ^{2n} implies that b = 2n.

To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(γ), Φ(δ)⟩ = ⟨(z, w) ↦ (z + 1, −w), (z, w) ↦ z, w + 1)⟩ ≅ ℤ˜×ℤ. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient Kb. Similarly

\overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)↦^{[α]}(x + 1, y), (x, y)↦^{[β]}(x, y + 2n)⟩ ≅ ℤ^{2} acts on ℝ^{2}⟨x, y⟩ with

quotient T^{2} so we have the desired representation.

The family of bundle-spaces (φ_{2}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) admit geometries of type E^{4}.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ^{−1}, δ), \\ [α, β] = γ^{(2)}, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & (2) & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨e_{1}, e_{2}+(1)e_{3}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ −(1) \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

52 ROBIN J. COBB

The subgroup A⟨α^{2}, β, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ −(1) \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{2}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) admits an E^{4} geometry. The bundle-space (φ_{3}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ^{−1}δ, δ), \\ [α, β] = γ^{(2)}, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & (2) & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨2e_{1} + e_{4}, e_{2} + (1)e_{3}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ −(1) \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \frac{1}{2} ; 0 ; 0 ; − \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}1 \\ 0 \\ 0 \\ −1 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ −(1) \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{3}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) admits an E^{4} geometry. The bundle-space (φ_{1}, φ_{2}; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γδ, δ), \\ [α, β] = 1, & β(γ, δ)β^{−1} = (γ^{−1}, δ). \end{matrix} ⟩ .

Modifying the geometric representation of (φ_{1}, e; ( \begin{matrix}0 \\ 0 \end{matrix} )) we obtain the representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \frac{1}{2} ; 0 ; 0 ; − \frac{1}{2} ) , I). \end{matrix}

The subgroup A⟨α, β^{2}, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \frac{1}{2} ; 0 ; 0 ; − \frac{1}{2} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{1}, φ_{2}; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

The bundle-space (φ_{1}, φ_{2}; ( \begin{matrix}2 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γδ, δ), \\ [α, β] = γ^{2}, & β(γ, δ)β^{−1} = (γ^{−1}, δ). \end{matrix} ⟩ .

Modifying the geometric representation of (φ_{1}, φ_{2}; ( \begin{matrix}0 \\ 0 \end{matrix} )) we obtain the representa-

tion Φ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ −4 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \frac{1}{2} ; 0 ; 0 ; − \frac{1}{2} ) , I). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨e_{1} −2e_{3}, e_{2}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \frac{1}{2} ; 0 ; 1 ; − \frac{1}{2} ) , I). \end{matrix}

The subgroup A⟨α, β^{2}, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \frac{1}{2} ; 0 ; 1 ; − \frac{1}{2} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{1}, φ_{2}; ( \begin{matrix}2 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Kb-bundles over Kb.

Here we fix the projection q to be:

q: Π_{1}(Kb) ↠ H_{1}(Kb) ↠ H_{1}(Kb)/Torsion ≅ ℤ⟨α⟩.

As before writing Π_{1}(M) = ker(qp) ⋊_{ψ} ℤ, we proceed in cases according to the image of ϑ: ℤ˜×ℤ → (ℤ/2ℤ)^{2}.

When β_{1}(M) = 1, ker(qp) = H is characteristic and we may use the following lemma to avoid redundancies in our listing of such bundle-spaces.

Lemma. Let Π = H ⋊_{θ} ℤ where the subgroup H ◁ Π is characteristic; then ϕ: Π → Π′ is an isomorphism precisely when Π′ = im(ϕ|_{H}) ⋊_{θ′} ℤ, where the images of θ and θ′ in the outer automorphism group of H are congugate via ϕ|_{H} up to inversion.

Proof.

Let Π and Π′ be groups as above with actions θ: ℤ → Aut(H) and θ′: ℤ → Aut(im(ϕ|_{H})) defined by respective cross-sections s: ℤ → Π and s′: ℤ → Π′. And let ϵ ∈ {±1} = Aut(ℤ) be the induced isomorphism on the quotient of Π by H as shown below:

\xymatrix{1 \ar[r] & H \ar[r] \ar[d]_{ϕ|_{H}} & Π \ar@{->>}[r]^{p} \ar[d]_{ϕ} & ℤ \ar[l]^{s} \ar[d]_{ϵ} \ar[r] & 1 \\ 1 \ar[r] & ϕ(H) \ar[r] & Π′ \ar@{->>}[r]^{p′} & ℤ \ar[l]^{s′} \ar[r] & 1}

Now ϕ^{−1}s_{2}ϵ is also a cross-section to p; thus writing ∗ to indicate multiplication in Π we have s^{−1}(n) ∗ ϕ^{−1}s′ϵ(n) ∈ ker(p) = H for all n ∈ ℤ. That is ϕs(n) = s′ϵ(n) ∗ h_{0} for some h_{0} ∈ H. Hence for any h ∈ H applying ϕ to the element θ(n)(h) := s(n) ∗ h ∗ s^{−1}(h) yields the equation

ϕ|_{H}θ(n)(h) = ϕs(n) ∗ ϕ(h) ∗ (ϕs(n))^{−1}
= s′ϵ(n) ∗ h_{0} ∗ ϕ|_{H}(h) ∗ h_{0}^{−1} ∗ ϵ^{−1}s′^{−1}(n)
= θ′(ϵ(n))(h_{0} ∗ ϕ|_{H}(h) ∗ h_{0}^{−1})
= θ′(ϵ(n))(C_{h_{0}}(ϕ|_{H}(h))
= (θ′)^{ϵ}C_{h_{0}}(ϕ|_{H}(h));

where C_{h_{0}}: H → H denote conjugation by h_{0}. Thus passing to outer automorphisms we have that [θ(n)] = (ϕ|_{H})^{−1}[(θ′)^{ϵ}(n)]ϕ|_{H} completing the proof. □

Remark. When Out(H) ≅ (ℤ/2ℤ)^{k} for some k ∈ ℕ, the groups Π and Π′ are isomorphic precisely when θ = θ′

The action of Π_{1}(Kb) on Π_{1}(Kb) is trivial.

In this case there only two extension equivalence classes since H^{2}(ℤ˜×ℤ; ℤ) ≅ ℤ/2ℤ.

Using the same argument as above when S = T^{2} with the exception that a = −1, this case yields the family of representations: (e, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℤ}. However by applying an appropriate change of cross-section of the form (α, β) ↦ (α, β)γ^{±2k}) for some k ∈ ℤ we see that this family of bundle-spaces represents just two bundlediffeomorphism classes given by the triples:

(e, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )).

These two triple represent distinct manifolds. This may be seen by calculating the characteristic subgroup I(Π′) = {g ∈ Π_{1}(M)| g^{n} ∈ [Π_{1}(M), Π_{1}(M)], ∃n ∈ ℕ \ {0}.} of Π_{1}(M) known as the isolator of the commutator subgroup. Here we have I(Π′) = ⟨β^{−1}γ^{(1)}, δ⟩ which is isomorphic to ℤ^{2} or the Klein bottle group depending on whether (1) = 0 or 1 respectively.

The action of Π_{1}(Kb) on Π_{1}(Kb) is via a non-trival proper subgroup of Out(Π_{1}(Kb)).

In this case im(ϑ) = ⟨φ_{i}⟩ for some fixed i ∈ {1, 2, 3}, and we may view this action as an epimorphism ϑ: ℤ˜×ℤ ↠ ℤ/2ℤ. If both generators α and β of ℤ˜×ℤ have non-trivial image under ϑ then we may apply an automorphism to ℤ˜×ℤ fixing β and mapping α to αβ. Therefore up to change of base of the base-space we may assume that either α or β is in the kernel of ϑ.

If β ∈ ker(ϑ) then θ = im(ϑ|_{⟨β⟩}) is trivial and ker(qp) ≅ B_{1}. Thus Π_{1}(M) ≅ B_{1} ⋊_{ψ} ℤ where ψ|_{⟨γ,δ⟩} = φ_{i} = im(ϑ). Examining the outer automorphisms of B_{1} we see that by taking c = 0, a = −1 and b an arbitrary even integer we are able to realize ψ for all values of i. That is we obtain the three families of representations:

(φ_{1}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℤ}, (φ_{2}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℤ}, (φ_{3}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℤ}. Counting equivalence classes,

the respective values of H^{2}(ℤ˜×ℤ; ℤ) for the above three families of triples are ℤ/2ℤ,ℤ or ℤ. In all of these cases by replacing γ by γ^{−1} if necessary, we may assume that n ∈ ℕ. However in the case where im(ϑ|_{⟨α⟩}) = φ_{1}; applying a change of cross-section of the form (α, β) ↦ (α, βγ^{2}k) for an appropriate k ∈ ℤ permits us to assume that n = (2). That is we obtain a pair and two infinite families of characteristic triples:

(φ_{1}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )), (φ_{2}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℕ}, (φ_{3}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℕ}.

Here the two groups represented by (φ_{1}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) are non-isomorphic and therefore represent distinct manifolds. This may be seen by calculating the isolator or the commutator subgroup. We have I(Π′) = ⟨β^{−1}γ^{(1)}, δ⟩ being isomorphic to ℤ^{2} or ℤ˜×ℤ depending on whether (1) = 0 or 1 respectively. The family of triples (φ_{2}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℕ} may be seen to represent a corresponding family of distinct bundle-spaces as follows. The kernel K ◁ Π_{1}(M) of the composite of qp followed the canonical projection:

\xymatrix{Π_{1}(M) \ar@{->>}[r] & H_{1}(M) \ar@{->>}[r] & H_{1}(M)/Torsion ≅ ℤ \ar@{->>}[r] & ℤ/2ℤ}

Here K ◁ Π_{1}(M) is characteristic as is ⟨δ⟩ ◁ Π_{1}(M), since

⟨δ⟩ = I([I([Π_{1}(M), Π_{1}(M)]), I([Π_{1}(M), Π_{1}(M)])).

Thus ⟨δ⟩ is characteristic in K and any automorphism of Π_{1}(M) induces an automorphism of K/⟨δ⟩. This quotient group K/⟨δ⟩ abelianizes to ℤ/4nℤ showing that n is indeed an invariant of this family of bundle-space representations. The same argument applied to the family (φ_{3}, e; ( \begin{matrix}n \\ 0 \end{matrix} ))_{n∈2ℕ} yields the abelianization of the group K/⟨δ⟩ as ℤ/nℤ showing that this family of triples represent distinct bundle-spaces and is distinct from the previous family.

If α ∈ ker(ϑ) then θ = φ_{i} is non-trivial so that ker(qp) ≅ B_{i+1} and Π_{1}(M) ≅ B_{i+1} ⋊_{ψ} ℤ where ψ acts trivially on ⟨γ, δ⟩ < B_{i+1}. Thus the monodromies are determined and we need only calculate the possible Euler numbers.

(1) (i = 1).H^{2}(ℤ˜×ℤ; ℤ) ≅ ℤ/2ℤ In this case ker(qp) ≅ B_{2} and the outer automorphism group Out(B_{2}) is isomorphic to the subgroup Γ_{2} < GL(2, ℤ consisting of matrices congruent modulo 2 to lower triangular matrices.

More specifically the commutator subgroup [B_{2}, B_{2}] = ⟨δ⟩ of B_{2} is characteristic. Thus any automorphism ϕ of B_{2} restricts to an automorphism of ⟨δ⟩ ≅ ℤ. Moreover ϕ induces an automorphism on the quotient B_{2}/⟨δ⟩ ≅ ℤ^{2}; which may be represented by A = ( \begin{matrix}a & c \\ b & d \end{matrix} ) ∈ GL(2, ℤ), in the sense that

(β, γ, δ)⟼^{ϕ}(β^{a}γ^{b}δ^{r}, β^{c}γ^{d}δ^{s}, δ^{ϵ}),

for some r, s ∈ ℤ and ϵ = ±1. Now requiring that ϕ(β) commute with ϕ(δ) shows that A ∈ Γ_{2}.

Now automorphisms that act identically on B_{2}/⟨δ⟩ (i.e when a = 1 = d and b = 0 = c) are precisely the inner automorphisms of B_{2}. Thus Inn(B_{2}) is the kernel of the map on automorphisms induced by abelianization of B_{2}:

ab^{∗}: Aut(B_{2}) → Aut(B_{2}/⟨δ⟩).

Therefore Out(B_{2}) ≅ Γ_{2} is isomorphic to the image of ab^{∗}; and may be generated by the following family of representative automorphisms:

(β, γ, δ) ⟼ (β^{a}γ^{b}δ^{\frac{a−1}{2}} , β^{c}γ^{d}, δ).

Thus the only automorphisms (unique up to inner automorphisms) that fix ⟨γ, δ⟩ and invert β have a = −1, d = 1 and c = 0. Hence the Euler number is of the form ( \begin{matrix}n \\ −1 \end{matrix} ), where n is even. A change of cross-section of the form (α, β) ↦ (α, βγ^{2}) transforms n into (2), and we obtain the pair of representations:

(e, φ_{1}; ( \begin{matrix}0 \\ −1 \end{matrix} )), (e, φ_{1}; ( \begin{matrix}2 \\ −1 \end{matrix} )).

these two triples indeed represent distinct bundle-spaces as H_{1}(M) is isomorphic to ℤ^{2} ⊕ ℤ/2ℤ when (2) = 0 and ℤ^{2} otherwise.

(2) (i = 2).H^{2}(ℤ˜×ℤ; ℤ) ≅ ℤ/2ℤ Here we see that the only outer automorphisms of B_{3} that fix each element of ⟨γ, δ⟩ and invert β are μ_{1} and μ_{1}μ_{3}. Thus noting that βγ^{2} = γ^{−2}β we may apply an automorphism to the fibre that inverts γ so as to obtain the two representations:

(e, φ_{2}; ( \begin{matrix}0 \\ 0 \end{matrix} )), (e, φ_{2}; ( \begin{matrix}2 \\ 0 \end{matrix} )).

These bundle-spaces have Π_{1}(M) ≅ B_{3} ⋊ ℤ where Out(B_{3}) ≅ (ℤ/2ℤ)^{3}. Since μ_{1} and μ_{1}μ_{2} are distinct elements of Out(B_{3}), by the above lemma these triples represent distinct bundle-spaces.

(3) (i = 3).H^{2}(ℤ˜×ℤ; ℤ) ≅ ℤ/2ℤ Here the group Out(B_{4}) ≅ (ℤ/2ℤ)^{3} and is generated by the automorphisms:

(β, γ, δ)⟼^{ν_{1}}(β^{−1}δ^{−1}, γ, δ), (β, γ, δ)⟼^{ν_{2}}(β, γ^{−1}, δ), (β, γ, δ)⟼^{ν_{3}}(βγ^{2}, γ, δ).

Remark The outer automorphism ν_{1} is incorrectly stated in [H1] Of these, ν_{1} and ν_{1}ν_{3} are the only automorphisms of B_{4} that act identically on the subgroup ⟨γ, δ⟩ and invert β. Thus we obtain the two representations:

(e, φ_{3}; ( \begin{matrix}0 \\ −1 \end{matrix} )), (e, φ_{3}; ( \begin{matrix}2 \\ −1 \end{matrix} )).

These two triples are shown to represent distinct bundle-spaces by abelianizing to obtain: H_{1}(M) ≅ ℤ⊕ℤ/2ℤ⊕(ℤ/4ℤ)^{2} or H_{1}(M) ≅ ℤ⊕(ℤ/2ℤ)^{2}⊕ℤ/4ℤ respectively.

The action of Π_{1}(Kb) on Π_{1}(Kb) is via the whole group Out(Π_{1}(Kb)).

In this case, up to equivalence there are three distinct epimorphisms ϑ: ℤ˜×ℤ ↠ (ℤ/2ℤ)^{2} distinguished by the image of β. This is because the subgroup ⟨β⟩ is characteristic in ℤ˜×ℤ and the image of α under ϑ may be altered by an automorphism mapping α to αβ. Therefore a change of base of the base-space defines an equivalence of such epimorphisms, partioned into classes characterized by the image of β. We choose representatives for these monodromies (ϑ(α), ϑ(β)) to be:

(φ_{2}, φ_{1}), (φ_{1}, φ_{2}), (φ_{1}, φ_{3}),

and proceed according to these three respective cases to determine the possible Euler numbers. Note that the respective extension equivalence classes corresponding to these actions have cardinalities in bijection with H^{2}(ℤ˜×ℤ; ℤ) ≅ ℤ, ℤ/2ℤ or ℤ/2ℤ respectively.

(1) In this case ker(qp) ≅ B_{2} and therfore ψ ∈ Out(B_{2}) must be of the form:

(β, γ, δ)⟼^{ψ}(γ^{b}δ^{−1}β^{−1}, γ^{−1}, δ),

where b is even. Hence by inverting, γ if necessary ,we may assume that the Euler numbers are of the form ( \begin{matrix}n \\ −1 \end{matrix} ) where n ∈ ℕ is even. That is we obtain the family:

(φ_{2}, φ_{1}; ( \begin{matrix}n \\ −1 \end{matrix} ))_{n∈2ℕ}.

As before the kernel K of the below projection:

\xymatrix{Π_{1}(M) \ar@{->>}[r] & H_{1}(M) \ar@{->>}[r] & H_{1}(M)/Torsion ≅ ℤ \ar@{->>}[r] & ℤ/2ℤ}

is characteristic. Similarly ⟨δ⟩ = [I([Π_{1}(M), Π_{1}(M)]), I([Π_{1}(M), Π_{1}(M)] and is therefore characteristic in Π_{1}(M) and hence in K. Abelianization of the quotient of K by the characteristic subgroup ⟨δ⟩ yields ℤ^{2} ⊕ ℤ/4nℤ. Therefore the above family of triples is in correspondence with a distinct family of bundle-spaces.

(2) In this case ker(qp) ≅ B_{3} So that the possibilities for ψ ∈ Out(B_{3}) are μ_{1}μ_{2} and μ_{1}μ_{2}μ_{3}. By inverting γ we may change the sign of the Euler number, so that we obtain the two representations:

(φ_{1}, φ_{2}; ( \begin{matrix}0 \\ 0 \end{matrix} )), (φ_{1}, φ_{2}; ( \begin{matrix}2 \\ 0 \end{matrix} )).

These bundle-spaces have Π_{1}(M) ≅ B_{3} ⋊ ℤ where Out(B_{3}) ≅ (ℤ/2ℤ)^{3}. Since μ_{1}μ_{2} and μ_{1}μ_{2}μ_{3} are distinct elements of Out(B_{3}), by the above lemma these triples represent distinct bundle-spaces.

(3) In this case ker(qp) ≅ B_{4}. Changing the representative of ν_{2} ∈ Out(B_{4}) to

(β, γ, δ) ⟼^{ν̂_{2}=ς_{β}ν_{2}}(β, γδ, δ)

where ς_{β} ∈ Inn(B_{4}) denotes conjugation by β; we see that the only possibilities for ψ are ν_{1}ν̂_{2} and ν_{1}ν̂_{2}ν_{3}. Thus we obtain the two representations:

(φ_{1}, φ_{3}; ( \begin{matrix}0 \\ −1 \end{matrix} )), (φ_{1}, φ_{3}; ( \begin{matrix}2 \\ −1 \end{matrix} )).

Here Out(B_{4}) ≅ (ℤ/2ℤ)^{3}. Since ν_{1}ν̂_{2} and ν_{1}ν̂_{2}ν_{3} are distinct elements of Out(B_{4}), by the above lemma these triples represent distinct bundle-spaces.

We now show that these bundle-spaces admit geometries as claimed.

The bundle-spaces (e, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) admit geometries of type E^{4}.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ, δ), \\ αβα^{−1}β = γ^{(2)}, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & (2) & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨e_{1}, e_{2}−(1)e_{3}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ (1) \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ (1) \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (e, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) admits an E^{4} geometry.

The bundle-spaces (φ_{1}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) admit geometries of type E^{4}.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γδ, δ), \\ αβα^{−1}β = γ^{(2)}, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( 0 \frac{1}{2} 0 0 ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ −1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 \frac{(1)}{2} 1 0 ) , I), \\ α & ↦ (( \frac{1}{2} 0 0 \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}1 \\ 0 \\ 0 \\ 1 \end{matrix} ) , ( 0 \frac{(1)}{2} 1 0 ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ −1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{1}, e; ( \begin{matrix}(2) \\ 0 \end{matrix} )) admits an E^{4} geometry.

The bundle-spaces (e, φ_{1}; ( \begin{matrix}(2) \\ −1 \end{matrix} )) admit geometries of type E^{4}.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ, δ), \\ αβα^{−1}β = γ^{(2)}δ^{−1}, & β(γ, δ)β^{−1} = (γδ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( 0 0 \frac{1}{2} 0 ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 ; \frac{1}{2} ; \frac{(1)}{2} ; − \frac{1}{2} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( 0 ; \frac{1}{2} ; \frac{(1)}{2} ; − \frac{1}{2} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (e, φ_{1}; ( \begin{matrix}(2) \\ −1 \end{matrix} )) admits an E^{4} geometry.

The bundle-space (φ_{2}, e; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ^{−1}, δ), \\ αβα^{−1}β = 1, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as follows:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, γ^{2}, δ⟩ of Π_{1}(M) has finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ n \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{2}, e; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

The family of bundle-spaces (φ_{2}, e; ( \begin{matrix}2n \\ 0 \end{matrix} ))_{n∈ℕ\\{0\}} admit geometries of type

Nil^{3} × E.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ^{−1}, δ), \\ αβα^{−1}β = γ^{2n}, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(γ)(x, y, z, w) = (x, y, z + 1, −w),
i.e Φ(γ) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯)).
Φ(δ)(x, y, z, w) = (x, y, z, w + 1)
i.e Φ(δ) = (I, (1, 1)).
Φ(β)(x, y, z, w) = (x, y + b, z, w)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(α)(x, y, z, w) = (x + 1, −y, −z − y, w)
i.e Φ(α) = (( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1))ρ;

where b is non-zero and ρ is the automorphism (x, y, z, w) ↦ (x, −y, −z, w).

Then the relations γδγ^{−1}δ = 1, α(γ, δ)α^{−1} = (γ^{−1}, δ) and β(γ, δ)β^{−1} = (γ, δ) are preserved by Φ. The remaining relation αβα^{−1}β = γ^{2n} implies that b = −2n.

To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(γ), Φ(δ)⟩ = ⟨(z, w) ↦ (z + 1, −w), (z, w) ↦ z, w + 1)⟩ ≅ ℤ˜×ℤ. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient Kb. Similarly

\overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)↦^{[α]}(x + 1, −y), (x, y)↦^{[β]}(x, y − 2n)⟩ ≅ ℤ˜×ℤ acts on ℝ^{2}⟨x, y⟩

with quotient Kb so we have the desired representation.

ON FIBRE-BUNDLES WITH BASE AND FIBRE EUCLIDEAN SURFACES 61

The bundle-space (φ_{3}, e; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ^{−1}δ, δ), \\ αβα^{−1}β = 1, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

We may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨2e_{1} +e_{4}, e_{2}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , I), \\ α & ↦ (( \frac{1}{2} ; 0 ; 0 ; − \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, γ^{2}, δ⟩ of Π_{1}(M) has finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}1 \\ 0 \\ 0 \\ −1 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{3}, e; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry. The family of bundle-spaces (φ_{3}, e; ( \begin{matrix}2n \\ 0 \end{matrix} ))_{n∈ℕ\\{0\}} admit geometries of type Nil^{3} × E.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ^{−1}δ, δ), \\ αβα^{−1}β = γ^{2n}, & β(γ, δ)β^{−1} = (γ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(γ)(x, y, z, w) = (x, y, z + 1, −w),
i.e Φ(γ) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯)).
Φ(δ)(x, y, z, w) = (x, y, z, w + 1)
i.e Φ(δ) = (I, (1, 1)).
Φ(β)(x, y, z, w) = (x, y + b, z, w)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} ) , (0, 1)).
Φ(α)(x, y, z, w) = (x + 1, −y, −z − y, w + a)
i.e Φ(α) = (( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (a, 1))ρ;

where b is non-zero and ρ is the automorphism (x, y, z, w) ↦ (x, −y, −z, w).

Then the relations γδγ^{−1}δ = 1, αδα^{−1} = δ and β(γ, δ)β^{−1} = (γ, δ) are preserved by Φ. The relation αγα^{−1} = γ^{−1}δ implies that a = −1/2 and the remaining relation αβα^{−1}β = γ^{2n} gives b = −2n.

To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(γ), Φ(δ)⟩ = ⟨(z, w) ↦ (z + 1, −w), (z, w) ↦ z, w + 1)⟩ ≅ ℤ˜×ℤ. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient Kb. Similarly

\overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)↦^{[α]}(x + 1, −y), (x, y)↦^{[β]}(x, y − 2n)⟩ ≅ ℤ˜×ℤ acts on ℝ^{2}⟨x, y⟩

with quotient Kb so we have the desired representation.

The bundle-spaces (e, φ_{1}; ( \begin{matrix}(2) \\ −1 \end{matrix} )) admit geometries of type E^{4}.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ, δ), \\ αβα^{−1}β = γ^{(2)}δ^{−1}, & β(γ, δ)β^{−1} = (γδ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( 0 0 \frac{1}{2} 0 ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 ; \frac{1}{2} ; \frac{(1)}{2} ; − \frac{1}{2} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( 0 ; \frac{1}{2} ; \frac{(1)}{2} ; − \frac{1}{2} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (e, φ_{1}; ( \begin{matrix}(2) \\ −1 \end{matrix} )) admits an E^{4} geometry.

The bundle-spaces (e, φ_{2}; ( \begin{matrix}(2) \\ 0 \end{matrix} )) admit geometries of type E^{4}.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ, δ), \\ αβα^{−1}β = γ^{(2)}, & β(γ, δ)β^{−1} = (γ^{−1}, δ). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ −(2) & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )); \end{matrix}

If (2) = 0 the image of ϕ is contained in ℝ^{4} ⋊ O(4). Furthermore in this case the subgroup A⟨α^{2}, β^{2}, γ^{2}, δ⟩ of Π_{1}(M) has finite index and maps faithfully into ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}. Thus

this representation is both faithful and discrete with cocompact image so that (e, φ_{2}; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

If (2) = 2 we may change the ordered basis to ℝ^{4} = ⟨e_{1} − e_{3}, e_{2}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, γ^{2}, δ⟩ of Π_{1}(M) has finite index and maps faithfully

into ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (e, φ_{2}; ( \begin{matrix}2 \\ 0 \end{matrix} )) admits an E^{4} geometry.

The bundle-spaces (e, φ_{3}; ( \begin{matrix}(2) \\ −1 \end{matrix} )) admit geometries of type E^{4}.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ, δ), \\ αβα^{−1}β = γ^{(2)}δ^{−1}, & β(γ, δ)β^{−1} = (γ^{−1}δ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( 0 0 \frac{1}{2} \frac{3}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 ; \frac{1}{2} ; − \frac{1}{2} ; − \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ (1) \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ (2) \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ −1 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (e, φ_{3}; ( \begin{matrix}(2) \\ −1 \end{matrix} )) admits an E^{4} geometry.

The bundle-space (φ_{2}, φ_{1}; ( \begin{matrix}0 \\ −1 \end{matrix} )) admits an E^{4} geometry.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ^{−1}, δ), \\ αβα^{−1}β = δ^{−1}, & β(γ, δ)β^{−1} = (γδ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( 0 0 \frac{1}{2} 0 ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 ; \frac{1}{2} ; 0 ; − \frac{1}{2} ) , I), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( 0 ; \frac{1}{2} ; 0 ; − \frac{1}{2} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{2}, φ_{1}; ( \begin{matrix}0 \\ −1 \end{matrix} )) admits an E^{4} geometry.

The family of bundle-spaces (φ_{2}, φ_{1}; ( \begin{matrix}2n \\ −1 \end{matrix} ))_{n∈ℕ\\{0\}} admit geometries of

type Nil^{3} × E.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γ^{−1}, δ), \\ αβα^{−1}β = γ^{2n}δ^{−1}, & β(γ, δ)β^{−1} = (γδ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → I som(Nil^{3} × E) as follows:

Φ(γ)(x, y, z, w) = (x, y, z + 1, −w),
i.e Φ(γ) = (( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 𝔯)).
Φ(δ)(x, y, z, w) = (x, y, z, w + 1)
i.e Φ(δ) = (I, (1, 1)).
Φ(β)(x, y, z, w) = (x, y + b, z, w + k)
i.e Φ(β) = (( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} ) , (k, 1)).
Φ(α)(x, y, z, w) = (x + 1, −y, −z − y, w)
i.e Φ(α) = (( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , (0, 1))ρ;

where b is non-zero and ρ is the automorphism (x, y, z, w) ↦ (x, −y, −z, w).

Then the relations γδγ^{−1}δ = 1, α(γ, δ)α^{−1} = (γ^{−1}, δ) and βδβ^{−1} = δ are preserved by Φ. The relation βγβ^{−1} = γδ implies that k = −1/2 and the remaining relation αβα^{−1}β = γ^{2n}δ^{−1} gives b = −2n.

To see that Φ is faithful and discrete with cocompact image note that the stabilizer of {(x_{1}, y_{1}, z, w)} ≤ I som(Nil^{3} × E) in the image of Φ for any fixed x_{1}, y_{1} ∈ ℝ is the group Φ(K) = ⟨Φ(γ), Φ(δ)⟩ = ⟨(z, w) ↦ (z + 1, −w), (z, w) ↦ z, w + 1)⟩ ≅ ℤ˜×ℤ. Clearly K acts on ℝ^{2}⟨z, w⟩ with quotient Kb. Similarly

\overline{Φ}(Π_{1}(M)/K) = ⟨(x, y)↦^{[α]}(x + 1, −y), (x, y)↦^{[β]}(x, y − 2n)⟩ ≅ ℤ˜×ℤ acts on ℝ^{2}⟨x, y⟩

with quotient Kb so we have the desired representation.

The bundle-spaces (φ_{1}, φ_{2}; ( \begin{matrix}(2) \\ 0 \end{matrix} )) admit geometries of type E^{4}.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γδ, δ), \\ αβα^{−1}β = γ^{(2)}, & β(γ, δ)β^{−1} = (γ^{−1}, δ). \end{matrix} ⟩ .

We define a representation ϕ: Π_{1}(M) → ℝ^{4} ⋊ GL(4, ℤ) as follows:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ −(2) & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

If (2) = 0 then we may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨2e_{1} + e_{4}, e_{2}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \frac{1}{2} ; 0 ; 0 ; − \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully

into ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}1 \\ 0 \\ 0 \\ −1 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{1}, φ_{2}; ( \begin{matrix}0 \\ 0 \end{matrix} )) admits an E^{4} geometry.

If (2) = 2 then we may change the ordered basis of ℝ^{4} = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ to ℝ^{4} = ⟨e_{1} − e_{3} + e_{4}, e_{2}, e_{3}, e_{4}⟩ to obtain a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( \begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ 1 \\ −1 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, γ^{2}, δ⟩ of Π_{1}(M) has finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ 2 \\ −2 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{1}, φ_{2}; ( \begin{matrix}2 \\ 0 \end{matrix} )) admits an E^{4} geometry.

The bundle-spaces (φ_{1}, φ_{3}; ( \begin{matrix}(2) \\ −1 \end{matrix} )) admit geometries of type E^{4}.

Π_{1}(M) = ⟨α, β, γ, δ | \begin{matrix}γδγ^{−1}δ = 1, & α(γ, δ)α^{−1} = (γδ, δ), \\ αβα^{−1}β = γ^{(2)}δ^{−1}, & β(γ, δ)β^{−1} = (γ^{−1}δ, δ). \end{matrix} ⟩ .

We define a representation Φ: Π_{1}(M) → ℝ^{4} ⋊ O(4) as shown below:

\begin{matrix}γ & ↦ (( 0 0 \frac{1}{2} \frac{3}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )), \\ δ & ↦ (( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , I), \end{matrix}  \begin{matrix}β & ↦ (( 0 ; \frac{1}{2} ; − \frac{1}{2} ; − \frac{1}{2} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )), \\ α & ↦ (( \begin{matrix}1 \\ 0 \\ (1) \\ − \frac{1}{2} \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )). \end{matrix}

The subgroup A⟨α^{2}, β^{2}, γ^{2}, δ⟩ of Π_{1}(M) is of finite index and maps faithfully into

ℝ^{4} under Φ with discrete, cocompact image ⟨( \begin{matrix}2 \\ 0 \\ (2) \\ −1 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 0 \\ −1 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} )⟩ ≅ ℤ^{4}.

Thus this representation is both faithful and discrete with cocompact image so that (φ_{1}, φ_{3}; ( \begin{matrix}(2) \\ −1 \end{matrix} )) admits an E^{4} geometry.

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