Embedding Seifert manifolds in the 4-sphere

J.S. Crisp

17 November, 1994

Abstract

The Z/2Z-signature theorem is used to obtain a relative bound on the Euler invariant ε ∈ Q of any Seifert manifold M, over a nonorientable closed surface, which embeds in 4-space. This bound is sharp whenever the singular fibres of M are all odd, and occur in pairs of opposite sign, and in the case of genuine circle bundles we obtain a complete classification of those which embed in 4-space. The torsion linking form is computed explicity for a Seifert manifold M over a nonorientable closed surface, and conditions are found for this to be hyperbolic, thereby giving extra necessary conditions for embedding M in S^{4}. These results allow us to show, for instance, that there is (up to orientation) exactly one manifold fibred over the orbifold RP^{2}(2, 2) which embeds in S^{4}.

1. Introduction

This paper is concerned with the general question of whether a given 3manifold embeds in S^{4}. By an embedding we shall mean a locally flat topological embedding, unless otherwise stated. However, all of the embeddings constructed in Section 2 are in fact smooth, so that for instance Corollaries 1.2 and 1.3 hold equally well for smooth embeddings. Only orientable 3-manifolds embed in S^{4}, and here we consider specifically the class of orientable Seifert manifolds [S], focussing mainly on those which are fibred over a nonorientable closed surface. In what follows, we shall write P_{k} for the nonorientable closed surface of genus k; thus P_{1} = RP^{2} is the real projective plane, and P_{2} = Kl the kleinbottle.

We may describe an orientable Seifert manifold M by the un-normalised data M(F; (α_{1}, β_{1}), …, (α_{n}, β_{n})) (following [NR] or [JN]), where M is fibred over the surface F with a finite collection of singular fibres γ_{i}, for i = 1, .., n, of type (α_{i}, β_{i}) where α_{i}, β_{i} are coprime, and α_{i} ≥ 1 is the multiplicity of γ_{i}. Note that [JN] also defined a generalised Seifert (G.S.) manifold, in which they allowed fibres of type (0, 1) as well. These are fibres fixed pointwise under the circle action as described in [O, §1.6]. We shall use this definition only as notational convenience in the proofs of Proposition 2.2 and Claim 4.5.

There is a fibre preserving homeomorphism which respects orientations between any two Seifert manifolds exactly when their Seifert data are equivalent via a finite sequence of the following modifications: (I) Add or delete any pair (1, 0); (II) Replace each pair (α_{i}, β_{i}) with (α_{i}, β_{i} +c_{i}α_{i}) where the c_{i} are chosen such that Σ_{i=1}^{n}c_{i} = 0; (III) Permute the indices. Note that there are some manifolds which admit more than one Seifert structure. These consist of the Lens spaces, and the following two exceptional cases ([JN], [NR], [O]) :

\begin{aligned}M(RP^{2}; (α, β)) & ≅ M(S^{2}; (2, 1), (2, −1), (−β, α)) \\ and M(Kl; (1, 0)) & ≅ M(S^{2}; (2, 1), (2, 1), (2, −1), (2, −1)) . \end{aligned}

For any Seifert manifold M = M(F; (α_{1}, β_{1}), …, (α_{n}, β_{n})) define the gen-

eralised Euler invariant to be ε(M) = ∑_{i=1}^{n} β_{i}/α_{i} ∈ Q. Clearly ε(M) is invariant

under (I)–(III). Observe also that, by applications of (I)–(III), we may assume that a Seifert manifold M has a unique standard form M(F; S, (1, β_{n+1})) where S = {γ_{i}(α_{i}, β_{i}) | α_{i} > β_{i} ≥ 1, i = 1, 2, .., n} is the set of strictly singular fibres. In view of this, each Seifert manifold M is determined by its base surface F,

the set S and its generalised Euler invariant ε = ε(M) = β_{n+1} + ∑_{i=1}^{n} β_{i}/α_{i}.

When F = P_{k} is nonorientable we shall write M_{S}(k; ε) for such a manifold. If we reverse the orientation of a Seifert manifold, then this may be reflected in its presentation by a change in the sign of each singular fibre, so that −M(F; (α_{i}, β_{i})) ≅ M(F; (α_{i}, −β_{i})). This also simply changes the sign of ε(M) so that we may write −M_{S}(k; ε) ≅ M_{−S}(k; −ε) where −S has the obvious definition.

As a first criterion for embeddability we note that if a 3-manifold M embeds in S^{4} then its torsion linking form must be hyperbolic, and in particular the torsion subgroup τ(M) of H_{1}(M) must be a direct double A ⊕ A [KK]. In Section 3 we compute explicitly the torsion linking form for the Seifert manifolds M_{S}(k; ε) over nonorientable closed surfaces. It follows that, for the linking to be hyperbolic, the following conditions must be satisfied. Let S_{even} denote the subset of S of fibres with even multiplicity. Then S_{even} consists of an even number, 2b say, of fibres (Remark 3.2), and there is a fixed t such that their multiplicities may all be written α_{i} = 2^{t}s_{i}, with s_{i} odd (Proposition 3.6). In such cases we associate two constants t_{S} and b_{S} to the set S as follows. When S_{even} is empty (i.e: all fibres have odd multiplicity) write t_{S} = b_{S} = 0. Otherwise write t_{S} = t, and b_{S} = b − 1. In Section 4 we prove the following theorem.

1.1. Theorem : For a given collection S of strictly singular fibres suppose that M_{1} = M_{S}(k_{1}; ε_{1}) and M_{2} = M_{S}(k_{2}; ε_{2}) both embed in S^{4}. Then

ε_{1} − ε_{2} ∈ \{\frac{2K}{2^{t_{S}}} , \frac{2K − 4}{2^{t_{S}}}, \frac{2K − 8}{2^{t_{S}}}, …, \frac{−2K + 4}{2^{t_{S}}}, \frac{−2K}{2^{t_{S}}} \} ∩ Z ,

where K = 2^{t_{S}}(k_{1} + k_{2} + 4b_{S}) + 4b_{S}.

In particular, if we suppose that every fibre of S has odd multiplicity (so that t_{S} = b_{S} = 0), fix k, and let ε_{0} be the least rational value such that M_{S}(k; ε_{0}) embeds in S^{4}, then M_{S}(k; ε) embeds only if

ε ∈ {ε_{0}, ε_{0} + 4, ε_{0} + 8, …, ε_{0} + 4k}.

When moreover the strictly singular fibres appear in pairs of “opposite sign”, that is S = −S, we can be more explicit.

1.2. Corollary : Suppose that S = {γ_{i}^{+}(α_{i}, β_{i}), γ_{i}^{−}(α_{i}, α_{i} − β_{i}) | i = 1, .., n}, where each α_{i} is odd. Then M_{S}(k; ε) embeds in S^{4} if and only if

ε ∈ {2k, 2k − 4, …, −2k} .

Proof: Examples of embeddings for each value of ε in the given range are provided in the proof of Proposition 2.2. On the other hand, Theorem 1.1 implies that ε may take on at most k + 1 different values, so that the set {2k, 2k − 4, …, −2k} must exhaust the possibilities. •

Note that the simple fact that M embeds in S^{4} if and only if −M embeds is reflected in Corollary 1.2.

An orientable circle bundle over a closed surface F may be expressed as a Seifert manifold M = M(F; (1, e)) with no strictly singular fibres, where e is the Euler class classifying the bundle. The circle bundles (necessarily orientable) which embed in S^{4} are classified as follows:

1.3. Corollary : Let M be an S^{1}-bundle with Euler class e, over a closed surface F. Then M embeds in S^{4} if and only if either

(1) F is orientable and e = 0 or ±1; or,

(2) F = P_{k} is nonorientable and e ∈ {2k, 2k − 4, …, −2k}.

Proof: The case when F is nonorientable follows immediately from Corollary 1.2. In the case where F is orientable Proposition 2.4 states that M embeds (smoothly in fact) for e = 0 or ±1. For the converse, we need only check the torsion subgroup τ(M) of H_{1}(M), since

H_{1}(M) ≅ ⟨a_{i}, b_{i}(i = 1, .., g), h | eh = 0⟩ ≅ Z^{2g} ⊕ Z/eZ,

so that τ(M) is a direct double only if e = 0 or ±1. •

Finally, as a further application of Theorem 1.1, when S = {(2, 1), (2, 1)} (t_{S} = 1 and b_{S} = 0), and F = RP^{2} we have ε_{1} − ε_{2} ∈ {4, 2, 0, −2, −4}, and since S = −S, ε ∈ {2, 0, −2}. Thus the only manifolds fibred over the orbifold RP^{2}(2, 2) which may possibly embed in S^{4} are the manifolds ±M(RP^{2}; (2, 3), (2, 1)) and M(RP^{2}; (2, 1), (2, −1)). The last of these fails to have a hyperbolic torsion linking form, as we show in Claim 3.7, and hence does not embed. This is, in fact, the Hantzsche-Wendt flat manifold. However an embedding of M(RP^{2}; (2, 3), (2, 1)) may be obtained by considering 0-framed surgery on the 8_{2}^{2} link (using Rolfsen’s notation [R1]), as in Example 2.1.

The key ingredient in the proof of Theorem 1.1 is the use of the Atiyah-Singer-Hirzebruch signature theorem. When considering the embeddings of nonorientable closed surfaces in S^{4}, Massey [M] used the Z/2Z-signature theorem to determine the possible range of the Euler class of the normal bundle of any embedded surface. A modification of this argument allows the analagous result to be proved for embeddings of circle bundles over these surfaces (Corollary 1.3). This was shown to me by J.A. Hillman for the specific cases of bundles over RP^{2} and Kl. Note that this is a strengthening of Massey’s argument, since an embedded surface P_{k} with Euler class e implies the existence of an embedded S^{1}-bundle with the same Euler class (namely ∂N(P_{k})), but not conversely.

Theorem 1.1 extends this constraint on embeddings to the whole class of Seifert manifolds over nonorientable closed surfaces. The proof takes on two phases. In the first place, given M_{1} and M_{2}, we construct a 4-manifold with involution g such that, by the signature theorem, sign(g, Σ) = 2^{t_{S}−1}(ε_{1} − ε_{2}). In the second place we use some algebraic topology to obtain an inequality which restricts the possible values of this signature. This inequality appears to be of a similar nature to the inequality given by Gilmer and Livingston in [GL, 2.1] which puts a bound on the Casson-Gordon signature invariant of (a cyclic cover of) a rational homology sphere that embeds in S^{4}.

Aside from the fact that the main theorem is in any case true for this more general class, the choice to consider embeddings to be TOP locally flat (as opposed to smooth) was motivated by recent work of Hillman [H] where it is shown that in some cases the property of TOP locally flat embeddability is common to a whole class of Z[π]-homology equivalent 3-manifolds (e.g: homology spheres, which all embed topologically but not always smoothly).

Acknowledgement :

I would like to thank Dr J.A. Hillman, who has supervised me during this work, for his many helpful suggestions and encouragement throughout.

2. Constructing Embeddings

In this section we consider various ways to construct smooth embeddings of Seifert manifolds, and in particular show the existence of all the embeddings claimed in the introduction. The simplest example is the standard embedding of S^{3} into S^{4} where S^{4} ≅ D^{4} ∪_{S^{3}} D^{4}. We may consider the 3-sphere as Seifert fibred in many ways. Namely S^{3} ≅ L(1, 0) ≅ M(S^{2}; (a_{1}, b_{1}), (a_{2}, b_{2})), whenever a_{1}b_{2} + a_{2}b_{1} = 1. More generally, any homology sphere embeds topologically [FQ]. These include, for instance, the manifolds obtained by (−1/n)-surgery on the trefoil knot [R2], which are Seifert fibred over S^{2}. However, not all of these embed smoothly.

Smooth embeddings exist for manifolds which are obtained by 0-framed surgery on a link in S^{3} which is the union of two sublinks L_{1}, L_{2}, each of which is slice. The embedding is obtained from the standardly embedded 3-sphere by attaching, on one side of S^{3}, an embedded 2-handle to each component of L_{1}, and similarly, on the other side, to each component of L_{2}. (See also [GL, §4]).

2.1. Example : The manifold M(RP^{2}; (2, 3), (2, 1)) may be obtained by 0framed surgery on the 8_{2}^{2} link (using Rolfsen’s notation [R1]), and hence we may embed it in S^{4}. This manifold has a surgery description (A) as in the figure, which is shown via the Kirby-Rolfsen [R2] calculus to be equivalent to the one claimed (B).

(A)

HAND-DRAWN DIAGRAM TO BE INSERTED

(B)

Given Seifert manifolds M_{1} and M_{2}, we write M_{1}§M_{2} for the Seifert manifold constructed as follows. Take regular fibres f_{i} of M_{i}, (i = 1, 2), and let M_{i}^{(f_{i})} = M_{i} \ int(N(f_{i})). Then M_{1}§M_{2} = M_{1}^{(f_{1})} ∪_{h} M_{2}^{(f_{2})}, where h : ∂N(f_{2}) → ∂N(f_{1}) is an orientation reversing fibre preserving homeomorphism. This operation is well-defined and if M_{i} = M(F^{(i)}; (α_{j}^{(i)}, β_{j}^{(i)}); j = 1, ..., n^{(i)}), then M_{1}§M_{2} = M(F^{(1)}♯F^{(2)}; (α_{j}^{(1)}, β_{j}^{(1)}); j = 1, ..., n^{(1)}, (α_{l}^{(2)}, β_{l}^{(2)}); l = 1, ..., n^{(2)}). If the M_{i} have singular fibres γ_{i} of type (p, q) and (p, −q) respectively, then we have a well-defined h : ∂N(γ_{2}) → ∂N(γ_{1}) as above, and we may extend the above construction to define M_{1} §_{γ_{1}}M_{2}. This has Seifert data as before, except that the fibres γ_{1} and γ_{2} are eliminated. This ‘fibre-sum’ operation is clearly commutative and, if restricted to regular fibres, associative as well.

If M embeds in S^{4} then M has a two-sided product neighbourhood in S^{4}. Thus we may embed two or more parallel copies of M, fibred in exactly the same way. Take two such which are the boundary components of M × I in S^{4}. Given a fibre γ of M then cleary ∂(M^{(γ)} × I) ≅ M §_{γ}−M also embeds in S^{4}. In fact, if we begin with r+1 parallel copies of M and take fibres γ_{i} (i = 1, .., r) in M such that γ_{i} ≠ γ_{i+1} (i = 1, .., r − 1), then we may construct in this way an embedding of

M §_{γ_{1}}−M §_{−γ_{2}}M §_{γ_{3}} … §_{±γ_{r}}±M.

This construction clearly gives smooth embeddings if we begin with a smooth embedding of M. For example, given any coprime pair (α, β) we may choose a, b ∈ Z such that M = M(0; (α, β), (a, b)) is a 3-sphere and hence embeds smoothly. Thus M §_{(a,b)}−M ≅ M(S^{2}; (α, β), (α, −β)) also embeds smoothly in S^{4}. Similarly, if we write M = M(S^{2}; (k, 1), (1, 0)) (again a sphere), then M §−M § M ≅ M(S^{2}; (k, 1), (k, −1), (k, 1)) embeds smoothly in S^{4}. These last manifolds may also be embedded via a 0-framed surgery, since

M(S^{2}; (k, 1), (k, −1), (k, 1)) ≅  ≅  .

Finally, let M ↪ S^{4} be the smooth embedding of Example 2.1. Then M§_{(2,3)}−M ≅ M(Kl; (2, 1), (2, −1)) also embeds smoothly.

In general the boundary of the regular disk-bundle neighbourhood of any surface F smoothly embedded in S^{4} is a circle bundle, M(F; (1, e)), with Euler number e equal to that of the normal bundle to F in S^{4}. Massey [M] constructed embeddings of the nonorientable surfaces P_{k} for all e ∈ {2k, 2k − 4, …, −2k}, and thus we have smooth embeddings of M(P_{k}; (1, e)) for all these values of e. We now extend these examples within the class of Seifert manifolds.

2.2. Proposition : The following Seifert manifolds embed smoothly in S^{4} for any e ∈ {2k, 2k − 4, …, −2k} :

M(P_{k}; (α_{1}, β_{1}), (α_{1}, −β_{1}), …, (α_{n}, β_{n}), (α_{n}, −β_{n}), (1, e)) ,

where each α_{i} is ODD. Note that here S = −S, and ε(M) = e.

Proof : Consider the lens space L(α_{i}, β_{i}) which we may write as a G.S. fibred manifold [JN]

L(α_{i}, β_{i}) ≅ M(S^{2}; (α_{i}, β_{i}), (0, 1)) = L_{i} say .

When α_{i} is odd L_{i} is a branched cyclic cover of S^{3}, branched over a knot. In this case, by [Zee], punc L_{i} = L_{i} \int(D^{3}) embeds smoothly in S^{4} as the fibre of a twist-spun 2-knot K. Now let punc L_{i} be constructed from the G.S. fibration above by removing an open neighbourhood of a point on the fibre of type (0, 1). Thus

punc L_{i} ≅ L_{i}^{∗} ⋃_{f×I} D^{2} × I ,

where L_{i}^{∗} = M(D^{2}; (α_{i}, β_{i})) is Seifert fibred, f is a regular fibre, I ≅ [0, 1], and the identification maps ∂D^{2} to f. Now punc L_{i} has a product neighbourhood punc L_{i} × [−ϵ, ϵ] in the exterior X(K) of K in S^{4}, and this may be written

punc L_{i} × [−ϵ, ϵ] = (L_{i}^{∗} × [−ϵ, ϵ]) ⋃(D^{2} × I × [−ϵ, ϵ]) .

Thus the 3-manifold

∂(L_{i}^{∗} × [−ϵ, ϵ]) ≅ L_{i} §_{(0,1)}−L_{i}
≅ M(S^{2}; (α_{i}, β_{i}), (α_{i}, −β_{i}))
= M_{i} say ,

clearly embeds smoothly in S^{4}. Furthermore, this embedding has the property that there exists a 4-ball B_{i} = D^{2} × I × [−ϵ, ϵ] embedded in S^{4} such that M_{i} ∩ B_{i} = ∂D^{2} × I × [−ϵ, ϵ], and each circle ∂D^{2} × {r, s} is a regular fibre in M_{i}. Notice that the embeddings of M(P_{k}; (1, e)) (for e ∈ {2k, 2k−4, …, −2k}) due to Massey [Ma] also have this property, since in fact each one bounds a disk bundle.

Now given any two such manifolds M_{1} and M_{2} embedded with this property, we may construct an embedding of M_{1}§M_{2}, also with this property, by taking connected sums of the 4-spheres. Remove the ball \frac{1}{2}B_{i} = D^{2}×I ×[\frac{−ϵ}{2}, \frac{ϵ}{2}] from each sphere, and construct the sum by taking the union of the closures of the remaining components in the obvious way. Thus, by induction, we obtain a smooth embedding of M_{1} § M_{2} § … § M_{n} § M(P_{k}; (1, e)), which is exactly the desired manifold, for any e ∈ {2k, 2k − 4, …, −2k}. • Note incidentally that this construction gives embeddings of

M(F; (α_{i}, β_{i}), (α_{i}, −β_{i}); i = 1, .., n) ,

with each α_{i} odd, for F = S^{2} immediately, and, by the following claim, for any closed orientable surface. Let M(T_{g}) denote the product bundle T_{g} × S^{1}, where T_{g} denotes the closed orientable surface of genus g.

2.3. Claim : Given a (smooth) embedding of M = M(F; (α_{i}, β_{i})) then we may construct a (smooth) embedding of M§M(T_{g}) ≅ M(F ♯T_{g}; (α_{i}, β_{i})).

Proof: Let f be a regular fibre of M embedded in S^{4}, and write B for the 3-ball with “equatorial” circle C ⊂ ∂B. Then f has in S^{4} a neighbourhood B × f consistently fibred with M, so that N = M ∩ (B × f) is a neighbourhood of f in M with ∂N = C × f. Write S^{∗} = S^{4} \ int(B × f) and define M^{(f)} as before to be M \ int(N). Then we have an embedding ϕ_{M} : M^{(f)} ↪ S^{∗} such that the restriction ∂M^{(f)} ↪ ∂S^{∗} = ∂B × f is a fibre preserving map with image C × f. On the other hand, write T_{g}^{∗} = T_{g} \ int(D^{2}). Since there are embeddings T_{g}^{∗} ↪ B with Im(∂T_{g}^{∗}) = C, there exists some fibre preserving embedding ϕ_{T} : T_{g}^{∗} × f ↪ B × f with Im(∂T_{g}^{∗} × f) = C × f. Together ϕ_{M} and ϕ_{T} give an embedding of M§M(T_{g}) into S^{4} = S^{∗} ∪ (B × f).

Clearly, this argument holds equally well for smooth embeddings. •

As an example, given the standard embedding of S^{3} ≅ M(S^{2}; (1, e)), a bundle with Euler number e = ±1, we may also smoothly embed M(T_{g}; (1, e)) with e = ±1 for any orientable surface T_{g}.

2.4. Proposition : The circle bundle M(T_{g}; (1, e)) embeds smoothly in S^{4} for e = 0, ±1.

Proof : Apart from the case just mentioned, we see that M(T_{g}; (1, 0)) = T_{g} × S^{1} embeds smoothly in the neighbourhood D^{3} × S^{1} of any circle in S^{4}. Alternatively, we have a different embedding of T_{g} × S^{1} as the boundary of the product neighbourhood of T_{g} ↪ S^{3} ↪ S^{4}. •

3. The Torsion Subgroup τ(M)

If a 3-manifold M embeds in S^{4} it must have a hyperbolic torsion linking form, and in particular the torsion subgroup τ(M) of H_{1}(M) must be a direct double A ⊕ A [KK]. In this section we determine a direct sum decomposition of τ(M) for an arbitrary Seifert manifold over a nonorientable closed surface, and then proceed to compute the torsion linking explicitly.

3.1. Lemma : Given M = M(P_{k}; (α_{1}, β_{1}), …, (α_{n}, β_{n})), where each α_{i} is written 2^{t_{i}}s_{i} for s_{i} odd, we may suppose that the α_{i}’s are ordered such that t_{1} = t_{2} = … = t_{r} > t_{r+1} ≥ … ≥ t_{n} ≥ 0. Then, if t_{1} ≠ 0, the torsion subgroup of H_{1}(M) is given by

τ(M) = (⨁_{i=3}^{n} Z/α_{i}Z) ⊕ \{ \begin{matrix}Z/2α_{1}Z ⊕ Z/2α_{2}Z , \\ Z/4α_{1}Z ⊕ Z/α_{2}Z , \end{matrix}  if r is even, ; if r is odd,

while, if t_{1} = 0 (i.e: every α_{i} is odd), we have

τ(M) = (⨁_{i=1}^{n} Z/α_{i}Z) ⊕ \{ \begin{matrix}Z/2Z ⊕ Z/2Z , \\ Z/4Z , \end{matrix}  if Σβ_{i} is even, if Σβ_{i} is odd.

Note that, by inserting pairs (1, 0) if necesssary, we may assume that n ≥ 2.

Proof: The group H_{1}(M) has the following presentation [S], [O], [JN] :

⟨h, a_{1}, .., a_{k}, q_{1}, .., q_{n} | 2h = 0, α_{j}q_{j} + β_{j}h = 0, 2 ∑_{i=1}^{k} a_{i} + ∑_{i=1}^{n} q_{j} = 0⟩ , (♠)

where each generator has an explicit representative relating to the Seifert structure as described following Remark 3.2. Replacing a_{k} with a = a_{1}+a_{2}+…+a_{k} this becomes Z^{k−1} ⊕ τ where

τ = ⟨h, a, q_{j}(j = 1, .., n) | 2h = 0, α_{j}q_{j} + β_{j}h = 0, 2a + Σq_{j} = 0⟩ , (♣)

and as we shall see τ is in fact the torsion subgroup τ(M).

Suppose first of all that t_{1} > 0. In this case, we may assume (by a Seifert equivalence if necessary) that every β_{j} is odd, and so rewrite the appropriate relations in the above presentation as α_{j}q_{j} = h. Now, for j ≥ 2, replace q_{j} with q̄_{j} = q_{j} + \frac{α_{1}}{2^{t_{j}}} q_{1}, and replace a with

ā = a − (∑_{j=2}^{n} \frac{α_{1}}{2^{t_{j}+1}} )q_{1} − ∑_{j=3}^{n} \frac{(s_{j} − 1)}{2}q̄_{j} + \{ \begin{matrix}\frac{1}{2}q_{1} , \\ − \frac{(s_{2} − 1)}{2}q̄_{2} , \end{matrix}  if r is even, ; if r is odd.

It is easily checked that this last equation contains an integral multiple of q_{1} in each case. Now the presentation for τ reduces to one of the following,

τ_{even} ≅ ⟨h, ā, q_{1}, q̄_{2}, .., q̄_{n} | 2h = 0, α_{1}q_{1} = h, α_{j}q̄_{j} = 0, 2ā + q̄_{2} + ∑_{j=3}^{n} s_{j}q̄_{j} = 0⟩ ,

τ_{odd} ≅ ⟨h, ā, q_{1}, q̄_{2}, .., q̄_{n} | 2h = 0, α_{1}q_{1} = h, α_{j}q̄_{j} = 0, 2ā + q_{1} + ∑_{j=2}^{n} s_{j}q̄_{j} = 0⟩ ,

as r is even or odd respectively.

If r is even, replace q̄_{2} with ¯ q̄_{2} = q̄_{2} + ∑_{j=3}^{n} s_{j}q̄_{j}. With this, the relation α_{2}q̄_{2} = 0 may be replaced with α_{2} ¯q̄_{2} = 0 (note that α_{j} divides α_{2}s_{j} for each j ≥ 3), and τ may be written

τ_{even} ≅ ⟨h, ā, q_{1}, ¯ q̄_{2}, .., q̄_{n} | 2h = 0, α_{1}q_{1} = h, α_{2} ¯q̄_{2} = α_{j}q̄_{j} = 0, 2ā + ¯ q̄_{2} = 0⟩

≅ ⟨ā, q_{1}, q̄_{3}, .., q̄_{n} | 2α_{1}q_{1} = 0, 2α_{2}ā = 0, α_{j}q̄_{j} = 0 (j ≥ 3)⟩

≅ Z/2α_{1}Z ⊕ Z/2α_{2}Z ⊕ (⨁_{i=3}^{n} Z/α_{i}Z) .

If r is odd, replace q_{1} with q̄_{1} = q_{1} + ∑_{j=2}^{n} s_{j}q̄_{j}. With this, the relation α_{1}q_{1} = h may be replaced with α_{1}q̄_{1} = h, and τ may be written

τ_{odd} ≅ ⟨h, ā, q̄_{1}, q̄_{2}, .., q̄_{n} | 2h = 0, α_{1}q̄_{1} = h, α_{j}q̄_{j} = 0 (j ≥ 2), 2ā + q̄_{1} = 0⟩

≅ ⟨ā, q̄_{1}, q̄_{2}, .., q̄_{n} | 4α_{1}ā = 0, α_{j}q̄_{j} = 0 (j ≥ 2)⟩

≅ Z/4α_{1}Z ⊕ (⨁_{i=2}^{n} Z/α_{i}Z) .

Finally, suppose that t_{1} = 0 and every α_{j} is odd. We may assume that α_{j}q_{j} = h for j = 1, .., s, and 0 for j > s, where s ≡ ∑_{j=1}^{n} β_{j} (mod 2). Putting

q̄_{j} = \{ \begin{matrix}q_{j} + α_{1}q_{1} = q_{j} + h , \\ q_{j} , \end{matrix} \begin{matrix}for j = 2, ..s, \\ for j > s. \end{matrix}

ā = a − (∑_{j=2}^{s} \frac{α_{1}}{2} )q_{1} − ∑_{j=3}^{n} \frac{(s_{j} − 1)}{2}q̄_{j} + \{ \begin{matrix}\frac{1}{2}q_{1} , \\ − \frac{(s_{2} − 1)}{2}q̄_{2} , \end{matrix}  if s is even, ; if s is odd,

the rest of the argument follows exactly as in the preceding cases. • 3.2. Remark : Since τ(M) must be a direct double for M to embed in S^{4}, it follows from the previous Lemma 3.1 that M(P_{k}; (α_{1}, β_{1}), …, (α_{n}, β_{n})) will only embed if the prime power factors of the α_{i}’s occur in pairs and, when each multiplicity is odd, only if ∑_{i=1}^{n} β_{i} is even. By looking at the torsion linking we find that, in addition, the nonzero t_{i}’s must all be equal (Proposition 3.6), and when each multiplicity is odd we have the stronger condition that ∑_{i=1}^{n} α_{i}β_{i} ≡ 2k mod 4 (Lemma 3.5).

Let <, >: τ(M) × τ(M) → Q/Z denote the torsion linking form on τ(M) of the Seifert manifold M discussed above. We now compute this form on the original generators of the presentation (♣) which is derived from the following description of M.

Take a basepoint ∗ in the surface P_{k} and let σ_{1}, ..., σ_{k} be orientation reversing loops in P_{k} at ∗ (each corresponding to a factor RP^{2} in the connected sum). Also choose loops ξ_{1}, .., ξ_{n} at ∗ in P_{k} which bound disks N_{1}, .., N_{n} respectively meeting only at ∗. Orient and label all these loops so that they appear as in Figure 3.3(a) below, and from now on consider them all as 1-cycles.

HAND-DRAWN DIAGRAM TO BE INSERTED

(a) (b) 3.3. Figure

Removing the interiors of the neighbourhoods N_{i} write B = P_{k} \ ⋃_{i=1}^{n} intN_{i}.

The result is contractible to a bouquet of circles and thus there is a unique orientable S^{1}-bundle B ˜×S^{1} over B, with a section B ↪ B ˜×S^{1}. We may now think of the σ_{i} and ξ_{j} as 1-cycles in B ˜×S^{1}. Also let f denote the fibre over ∗ ∈ B also considered as a 1-cycle in B ˜×S^{1}. Now let N(γ_{i}), for i = 1, .., n, be solid tori with cores γ_{i}, and write

M = B ˜×S^{1} ∪_{h_{i}} (⋃_{i=1}^{n} N(γ_{i})) ,

where each attaching homeomorphism h_{i} identifies the boundary of a meridinal disk in N(γ_{i}) with a curve representing the homology class α_{i}[ξ_{i}] + β_{i}[f]. Then M is the Seifert manifold M(P_{k}; (α_{i}, β_{i})) with singular fibres γ_{i}, and H_{1}(M) has the presentation (♠) where h = [f], a_{i} = [σ_{i}], and q_{j} = [ξ_{j}], for i = 1, .., k and j = 1, .., n. Define the 1-chain σ = σ_{1} + σ_{2} + … + σ_{k}, so that, in (♣), a = a_{1} + a_{2} + … + a_{k} = [σ]. The relations in (♣) correspond to the presence of the following 2-chains in M. The union of fibres over σ_{1} may be considered as a 2-chain A (the image of an annulus) with ∂A = 2f. For each i = 1, .., n, we may assume that there is a 2-chain D_{i} in N(γ_{i}) with ∂D_{i} = α_{i}ξ_{i} + β_{i}f in M (thus D_{i} is homologous to a meridian disk). Finally, we may consider B as

a 2-chain in M with ∂B = 2σ + ∑_{i=1}^{n} ξ_{i}. Note that we have described each of

∂A, ∂B and the ∂D_{i} explicitly as 1-cycles. We may now compute the following linkings. Let M be as in Lemma 3.1, with generalised Euler invariant ε.

3.4. Lemma : The torsion linking form <, >: τ(M) × τ(M) → Q/Z , with respect to the generators of (♣), is given as follows:

< q_{i}, q_{j} >= \{ \begin{matrix}β_{i}/α_{i} , \\ 0 , \end{matrix} \begin{matrix}if i = j, \\ if i ≠ j, \end{matrix} \begin{aligned}< a, q_{i} > & = −β_{i}/2α_{i} , \\ < a, a > & = (2k + ε)/4 , \\ < q_{i}, h > & =< h, h >= 0 , \\ and < a, h > & = 1/2 . \end{aligned}

Proof : The 1-cycles corresponding to the generators are shown in Figure 3.3(a) with f directed out of the page, and we use the right hand rule to determine a positive intersection. We may perturb each ξ_{i} to a homologous 1-cycle ξ′_{i} just inside N(γ_{i}), and perturb σ to σ′ as illustrated in Figure 3.3(b). Then we have the following intersection numbers: ξ′_{i}.A = ξ′_{i}.B = 0 and

ξ′_{i}.D_{j} = \{ \begin{matrix}β_{i} , \\ 0 , \end{matrix} \begin{matrix}if i = j, \\ if i ≠ j, \end{matrix}

while σ′.D_{i} = −β_{i}, and σ′.A = −1, and σ′.B = k (by a choice of σ′, since by a different choice we may get σ′.B = −k). Now, for j = 1, .., n, let C_{j} = 2D_{j} − β_{j}A, so that ∂C_{j} = 2α_{j}ξ_{j}. Then ξ′_{i}.C_{j} = 2ξ′_{i}.D_{j} = δ_{ij}2β_{i}, and consequently < q_{i}, q_{j} >= δ_{ij} \frac{β_{i}}{α_{i}}. Also σ′.C_{i} = −2β_{i} + β_{i} = −β_{i}, so that < a, q_{i} >= \frac{−β_{i}}{2α_{i}}.

Now write C = ∑_{i=1}^{n} α̂_{i}C_{i}, where α̂_{i} = \frac{1}{α_{i}} ∏_{j=1}^{n} α_{j}, and let R = 2( ∏_{j=1}^{n} α_{j})B − C.

Then ∂R = 4( ∏_{j=1}^{n} α_{j})σ. Also

σ′.R = 2(∏_{j=1}^{n} α_{j})k + ∑_{i=1}^{n} α̂_{i}β_{i} = (∏_{j=1}^{n} α_{j})(2k + ε) .

Thus < a, a >= \frac{(2k+ε)}{4}. (Note that we may also use R to check that < q_{i}, a >= \frac{−β_{i}}{2α_{i}} =< a, q_{i} > which follows also by the symmetry of the form). Finally, ξ′_{i}.A = 0 implies that < q_{i}, h >= 0 for each i = 1, .., n, and σ′.A = −1 implies that < a, h >= −1/2 = 1/2, since ∂A = 2f. Also we may perturb f to f′ so that f′.A = 0. Thus < h, h >= 0. •

3.5. Lemma : When every multiplicity α_{i} is odd, then the torsion linking form for M(P_{k}; (α_{1}, β_{1}), …, (α_{n}, β_{n})) is given by the linking matrix

T = ( \begin{matrix}β_{1}/α_{1} & 0 & & & 0 & 0 \\ 0 & β_{2}/α_{2} & & & 0 & 0 \\ & & . \\ & & & . \\ 0 & 0 & & & 0 & 1/2 \\ 0 & 0 & & & 1/2 & (2k + η)/4 \end{matrix} ) ,

where η = ∑_{i=1}^{n} α_{i}β_{i}. Also, η is invariant (mod 4) under Seifert data equivalences, and if the linking T is hyperbolic we must have η ≡ 2k (mod 4).

Proof : From Lemma 3.4, the form is described by the matrix

( \begin{matrix}β_{1}/α_{1} & 0 & & & 0 & −β_{1}/2α_{1} \\ 0 & β_{2}/α_{2} & & & 0 & −β_{2}/2α_{2} \\ & & . \\ & & & . \\ 0 & 0 & & & 0 & 1/2 \\ −β_{1}/2α_{1} & −β_{2}/2α_{2} & & & 1/2 & (2k + ε)/4 \end{matrix} ) .

Changing the generators by substituting â = a + ∑_{i=1}^{n} \frac{(α_{i}+1)}{2}q_{i} (noting that each

α_{i} is odd) results in the matrix

( \begin{matrix}β_{1}/α_{1} & 0 & & & 0 & β_{1}/2 \\ 0 & β_{2}/α_{2} & & & 0 & β_{2}/2 \\ & & . \\ & & & . \\ 0 & 0 & & & 0 & 1/2 \\ β_{1}/2 & β_{2}/2 & & & 1/2 & x \end{matrix} ) .

where we have x = \frac{k}{2} + \frac{ε}{4} + 2 ∑_{i=1}^{n}(\frac{(α_{i}+1)}{2})(\frac{−β_{i}}{2α_{i}}) + ∑_{i=1}^{n}(\frac{(α_{i}+1)}{2})^{2}(\frac{β_{i}}{α_{i}}) = \frac{k}{2} + \frac{ε}{4} +

\frac{1}{4} ∑_{i=1}^{n}(α_{i}^{2} − 1)(\frac{β_{i}}{α_{i}}) = \frac{k}{2} + \frac{1}{4} ∑_{i=1}^{n} α_{i}β_{i} = \frac{(2k+η)}{4}. Now, a substitution q̂_{i} = q_{i} − β_{i}h,

for each i = 1, .., n, leaves the matrix T.

We can check that η is invariant under Seifert equivalence, since if β′_{1} = β_{1} + α_{1}, and β′_{2} = β_{2} − α_{2}, then (α′_{1}β′_{1} + α′_{2}β′_{2}) − (α_{1}β_{1} + α_{2}β_{2}) = α_{1}^{2} − α_{2}^{2} = (α_{1} + α_{2})(α_{1} − α_{2}) ≡ 0 (mod 4). Also, it is easily checked that the block

( \begin{matrix}0 & 1/2 \\ 1/2 & (2k + η)/4 \end{matrix} ) is hyperbolic if and only if 2k + η ≡ 0 (mod 4). •

In the case where the strictly singular fibres occur in opposite pairs (i.e: S = −S), as in Corollary 1.2, we can easily check that η ≡ ε (mod 4), and

moreover, since ( \begin{matrix}β_{i}/α_{i} & 0 \\ 0 & −β_{i}/α_{i} \end{matrix} ) ∼ ( \begin{matrix}0 & 1/α_{i} \\ 1/α_{i} & 0 \end{matrix} ) the linking is clearly

hyperbolic in this case if and only if ε ≡ 2k (mod 4).

When considering the form in the general case, one can show the following.

3.6. Proposition : The manifold M = M(P_{k}; (α_{1}, β_{1}), …, (α_{n}, β_{n})) where we write α_{i} = 2^{t_{i}}s_{i} for s_{i} odd, has a hyperbolic torsion linking form only if, for each i = 1, .., n, either t_{i} = t_{1} or t_{i} = 0. That is, all the nonzero t_{i} must be equal.

Proof: We change our set of generators to a larger one as follows. Let c_{i} = 2^{t_{i}}q_{i} and d_{i} = s_{i}q_{i} for each i = 1, .., n. Then the set {a, h, c_{i}, d_{i} : i = 1, .., n} generates τ(M) , and we have (mod Z)

< c_{i}, c_{j} > = 2^{t_{i}}2^{t_{j}} < q_{i}, q_{j} >= \{ \begin{matrix}2^{t_{i}}β_{i}/s_{i} , \\ 0 , \end{matrix} \begin{matrix}if i = j, \\ if i ≠ j. \end{matrix}
< c_{i}, d_{j} > = 2^{t_{i}}s_{j} < q_{i}, q_{j} >= 0
< d_{i}, d_{j} > = s_{i}s_{j} < q_{i}, q_{j} >= \{ \begin{matrix}s_{i}β_{i}/2^{t_{i}} , \\ 0 , \end{matrix} \begin{matrix}if i = j, \\ if i ≠ j. \end{matrix}

< c_{i}, a > = 2^{t_{i}}(\frac{−β_{i}}{2α_{i}}) = \frac{−β_{i}}{2s_{i}}
< d_{i}, a > = s_{i}(\frac{−β_{i}}{2α_{i}}) = \frac{−β_{i}}{2^{t_{i}+1}} .

Finally < c_{i}, h >=< d_{i}, h >=< h, h >= 0 and < a, h >= 1/2 as before. Since s_{i} is odd, we may put each < c_{i}, a > entry to zero as in Lemma 3.5. This operation otherwise only affects the < a, a > entry, and gives a linking matrix which is a block sum of the diagonal matrix ( < c_{i}, c_{j} >) with

( \begin{matrix}\frac{s_{1}β_{1}}{2^{t_{1}}} & 0 & & & 0 & \frac{−β_{1}}{2^{t_{1}+1}} \\ 0 & \frac{s_{2}β_{2}}{2^{t_{2}}} & & & 0 & \frac{−β_{2}}{2^{t_{2}+1}} \\ & & . \\ & & & . \\ 0 & 0 & & & 0 & 1/2 \\ \frac{−β_{1}}{2^{t_{1}+1}} & \frac{−β_{2}}{2^{t_{2}+1}} & & & 1/2 & x \end{matrix} ) .

Now suppose that contrary to the conclusion of the theorem there is some i ∈ {2, .., n} such that 0 < t_{i} < t_{1} = t. Then we wish to show that the above form is not hyperbolic. Note that in this case α_{1} , α_{i} are even, so that β_{1} , β_{i} are odd (as well as s_{1} , s_{i} being odd). Replace the generator d_{i} with d̄_{i} = d_{i} +2^{t−t_{i}}s_{i}d_{1} (or just d̄_{i} = s_{i}q̄_{i} for q̄_{i} as in the proof of Lemma 3.1). Then

< d̄_{i}, d̄_{i} > =< d_{i}, d_{i} > +(2^{t−t_{i}}s_{i})^{2} < d_{1}, d_{1} >= \frac{s_{i}β_{i} + 2^{t−t_{i}}s_{i}^{2}s_{1}β_{1}}{2^{t_{i}}}
= \frac{(odd)}{2^{t_{i}}} since t_{1} < t and s_{i}β_{i} is odd,
< d̄_{i}, d_{1} > = 2^{t−t_{i}}s_{i}(\frac{s_{1}β_{1}}{2^{t}}) = \frac{(odd)}{2^{t_{i}}} ,
< d̄_{i}, d_{j} > = 0  for j ≠ 1, i,
< d̄_{i}, a > = \frac{−β_{i}}{2^{t_{i}+1}} − 2^{t−t_{i}}s_{i}(\frac{β_{1}}{2^{t+1}} ) = \frac{−(β_{i} + s_{i}β_{1})}{2^{t_{i}+1}} = \frac{\frac{1}{2}(even)}{2^{t_{i}}} ,
< d̄_{i}, h > = 0 .

The previous block now becomes

( \begin{matrix}\frac{(odd)}{2^{t}} & 0 & \frac{(odd)}{2^{t_{i}}} & 0 & . & 0 & \frac{(odd)}{2^{t+1}} \\ 0 & . & 0 & & & & . \\ \frac{(odd)}{2^{t_{i}}} & 0 & \frac{(odd)}{2^{t_{i}}} & 0 & . & 0 & \frac{\frac{1}{2} (even)}{2^{t_{i}}} \\ 0 & & 0 & . & & & . \\ . & & . & & . & & . \\ 0 & & 0 & & & 0 & 1/2 \\ \frac{(odd)}{2^{t+1}} & . & \frac{\frac{1}{2} (even)}{2^{t_{i}}} & . & . & 1/2 & x \end{matrix} ) ,

which is equivalent (by conjugation) to the linking matrix

( \begin{matrix}\frac{(odd)}{2^{t}} & 0 & 0 & 0 & . & 0 & \frac{(odd)}{2^{t+1}} \\ 0 & . & 0 & & & & . \\ 0 & 0 & \frac{(odd)}{2^{t_{i}}} & 0 & . & 0 & 0 \\ 0 & & 0 & . & & & . \\ . & & . & & . & & . \\ 0 & & 0 & & & 0 & 1/2 \\ \frac{(odd)}{2^{t+1}} & . & 0 & . & . & 1/2 & x′ \end{matrix} ) .

But, by Theorem 0.1 of [KK], a hyperbolic linking on a 2-group can have no diagonal summand and hence the form above is not hyperbolic. Linkings on finite abelian groups form an abelian semigroup under block sum, and [KK, Theorem 0.1] gives a set of generators and relations for the semigroup of linkings on 2-groups. According to [KK] any hyperbolic linking on a 2-group may be

written in terms of these generators as ⨁_{i} E_{0}^{k_{i}} where E_{0}^{k} = ( \begin{matrix}0 & 2^{−k} \\ 2^{−k} & 0 \end{matrix} ). Also

we have the relations 2E_{0}^{k} = 2E_{1}^{k} for k ≥ 2, where E_{1}^{k} = ( \begin{matrix}2^{1−k} & 2^{−k} \\ 2^{−k} & 2^{1−k} \end{matrix} ). But

the other relations of [KK, Theorem 0.1] allow an E_{0}^{k} or E_{1}^{k} to be exchanged for diagonal summands A^{k}(n) = ( \frac{n}{2^{k}} ) only if some other diagonal summand is already present, which is impossible if we start with a hyperbolic linking. Finally, the linking we are considering above may be written A^{t_{i}}(odd) ⊕ Λ for some linking Λ which may also be expressed in terms of the standard generators. Consequently, no hyperbolic form is equivalent to this one. •

Note that the odd order torsion linking can be treated as follows. The n × n block diag(\frac{2^{t_{i}} β_{i}}{s_{i}}) is equivalent (as a linking form) to

B = ⨁_{p prime , r≥1} B(p^{r}) ,

such that each B(p^{r}) is given by the matrix diag(\frac{b_{1}(p^{r})}{p^{r}}, \frac{b_{2}(p^{r})}{p^{r}}, …, \frac{b_{n}(p^{r})}{p^{r}}), where b_{i}(p^{r}) = \frac{α_{i}β_{i}}{p^{r}} if p^{r} divides s_{i}, and is 0 otherwise. Thus there is one diagonal entry for each distinct prime power factor of each s_{i}. The linking B is hyperbolic if and only if each B(p^{r}) is hyperbolic. That is if and only if

B(p^{r}) ∼ ⨁_{i=1}^{m} ( \begin{matrix}0 & \frac{1}{p^{r}} \\ \frac{1}{p^{r}} & 0 \end{matrix} ) = ⨁_{i=1}^{m} ( \begin{matrix}\frac{1}{p^{r}} & 0 \\ 0 & \frac{−1}{p^{r}} \end{matrix} ) .

Note that since ( \frac{b}{p^{r}} ) ∼ ( \frac{m^{2}b}{p^{r}} ) we may assume that each b_{i}(p^{r}) is either 1 or a quadratic non-residue. Then by Wall’s presentation ([W], [KK]) of the semigroup of linkings on p-groups (for a fixed odd prime p) — with generators ( \frac{1}{p^{r}} ) and ( \frac{n(p)}{p^{r}} ) and relations ( \frac{1}{p^{r}} ) ⊕ ( \frac{1}{p^{r}} ) = ( \frac{n(p)}{p^{r}} ) ⊕ ( \frac{n(p)}{p^{r}} ) for each n(p) a quadratic non-residue (mod p) — it is a routine matter to check whether or not the linking B(p^{r}) is hyperbolic. Thus in the case of Lemma 3.5 it is possible to give necessary and sufficient conditions for hyperbolicity.

In view of Proposition 3.6, we may consider the case where t = 1, in which case we have the following partial result :

3.7. Claim : With S = {γ_{i}(2, 1) | i = 1, .., 2m}, the manifold M_{S}(k; ε) has a hyperbolic torsion linking form if and only if δ = 2k + ε ≡ 0 (mod 4).

Proof : We consider only m = 1. The general case follows by the same argument applied m times. Note that, for the given S, ε and hence δ must be an integer. By Lemma 3.4, the torsion linking form has the following matrix:

( \begin{matrix}1/2 & 0 & 0 & −1/4 \\ 0 & 1/2 & 0 & −1/4 \\ 0 & 0 & 0 & 1/2 \\ −1/4 & −1/4 & 1/2 & δ/4 \end{matrix} ) ∼ ( \begin{matrix}\frac{1}{2} & 0 & \frac{1}{4} \\ 0 & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{δ}{4} \end{matrix} ) .

In what follows we use the notation (i) : +r(j) to mean replace the i^{t}h generator, g_{i} say, with g_{i} + rg_{j}. That is add r × row j to row i and r × col j to col i. When δ ≡ 0 (mod 4) this linking is equivalent to

( \begin{matrix}\frac{1}{2} & 0 & \frac{1}{4} \\ 0 & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & 0 \end{matrix} ) ∼^{(1):+(3)} ( \begin{matrix}0 & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & 0 \end{matrix} ) ∼^{(2):−(1)} ( \begin{matrix}0 & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & 0 & 0 \\ \frac{1}{4} & 0 & 0 \end{matrix} ) ∼ ( \begin{matrix}0 & \frac{1}{4} & 0 \\ \frac{1}{4} & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

If δ ≡ 1 (mod 4) then we have

( \begin{matrix}\frac{1}{2} & 0 & \frac{1}{4} \\ 0 & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{matrix} ) ∼^{(2):−(3)} ( \begin{matrix}\frac{1}{2} & \frac{−1}{4} & \frac{1}{4} \\ \frac{−1}{4} & \frac{1}{4} & 0 \\ \frac{1}{4} & 0 & \frac{1}{4} \end{matrix} ) ∼^{(1):−(3)} ( \begin{matrix}\frac{1}{4} & \frac{−1}{4} & 0 \\ \frac{−1}{4} & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{4} \end{matrix} ) ,

which fails to be hyperbolic, and similarly for δ ≡ −1 (mod 4). Finally when δ ≡ 2 (mod 4) we have

( \begin{matrix}\frac{1}{2} & 0 & \frac{1}{4} \\ 0 & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{2} \end{matrix} ) ∼^{(1):−(2)} ( \begin{matrix}0 & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{4} \\ 0 & \frac{1}{4} & \frac{1}{2} \end{matrix} ) ∼^{(1):+2(3)} ( \begin{matrix}0 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{1}{4} \\ 0 & \frac{1}{4} & \frac{1}{2} \end{matrix} ) ,

which again is not hyperbolic. •

In particular, the manifolds fibred over the orbifold RP^{2}(2, 2), that is the M(RP^{2}; (2, 1), (2, −1), (1, e)), have hyperbolic torsion linking if and only if e ≡ 2 (mod 4). Thus M(RP^{2}; (2, 1), (2, −1)) does not embed in S^{4}.

4. Proof of Theorem 1.1.

In what follows let M = M_{S}(k; ε), and suppose that M is embedded in the 4-sphere. Then M separates S^{4}, and we may write S^{4} = X ∪_{M} Y for some 4-manifolds X and Y. Note also that, by the Mayer-Vietoris sequence, H_{1}(M) ≅ H_{1}(X)⊕H_{1}(Y). With the notation of Lemma 3.1, the set S consists of fibres of multiplicity α_{i} = 2^{t_{i}}s_{i} for s_{i} odd, i = 1, .., n, and we may suppose, by Proposition 3.6, that t^{1} = t^{2} = .. = t^{r} = t_{S} while t^{r+1} = .. = t^{n} = 0. In what follows, we simply write t_{S} = t_{1} = t.

4.1 Lemma : Take M as above. Then for one of X or Y, say X without loss of generality, there exists a covering ψ_{X} : X̃ → X (which restricts to a covering ψ_{M} : M̃ → M on the boundary) together with a covering involution (of X̃) which restricts on M̃ to a rotation of the fibres through π radians. Here M̃ has the Seifert structure induced by ψ_{M}.

Proof : Since τ(M) must be a direct double, we may assume by Lemma 3.1, that it has a direct summand Z/2α_{1}Z generated by q_{1}, as in the proof. In fact q̂ = s_{1}q_{1} generates a direct summand of τ(M) of order 2^{t+1}. Thus we may define a surjective homomorphism

ϕ : H_{1}(M) ⟶ ⟨ζ | 2^{t+1}ζ = 0⟩ ≅ Z/2^{t+1}Z

such that, other than q̂ every generator of H_{1}(M) maps to 0, while q̂ ↦ ζ. This map is effectively a projection onto the summand of τ(M) generated by q̂.

Now q̂ may be identified with an element (a, b) ∈ H_{1}(X) ⊕ H_{1}(Y) ≅ H_{1}(M), and one of (a, 0) or (0, b), without loss of generality (a, 0) say, must have order 2^{t+1} not just in H_{1}(M), but also under the quotient map ϕ as well. In this case the homomorphism

α : H_{1}(X) ↪ H_{1}(M) ⟶^{ϕ} Z/2^{t+1}Z ,

such that x ↦ ϕ((x, 0)), must be surjective, since the image of a has order 2^{t+1} and so generates Z/2^{t+1}Z. Thus, taking

α_{π} : π_{1}(X) ⟶^{ab} H_{1}(X) ⟶^{α} Z/2^{t+1}Z ,

there exists a (2^{t+1})-fold covering ψ_{X} : X̃ → X with π_{1}(X̃) = ker α_{π}.

Let h ∈ H_{1}(M) be the element of order 2 represented by a regular fibre f of M. As in the proof of Lemma 3.1, h = α_{1}q_{1} = 2^{t}q̂. Since the inclusion M ⊂ X induces the projection p_{X} : H_{1}(M) → H_{1}(X), the fibre f of M represents in H_{1}(X) the element [f] = p_{X}(h) = p_{X}(2^{t}q̂) = 2^{t}a, and since α(a) has order 2^{t+1}, α([f]) will have order 2 in Z/2^{t+1}Z. Thus the loop f.f lifts to a loop in M̃ ([f]^{2} ∈ ker α_{π}), while f does not, which is to say that, for each regular fibre, ψ_{M}^{−1}(f) is a collection of circles each of which covers f two-fold.

On the other hand, if we take the map

μ : π_{1}(X) ⟶^{α_{π}} Z/2^{t+1}Z → Z/2^{t}Z ,

such that [f] ↦ 0, then we get an intermediate covering ψ̄_{X} : X̄ → X where each fibre of M lifts to a fibre in M̄. Thus X̃ is a 2-fold cover of X̄ with covering automorphism as required. That is each regular fibre of M̃ covers a fibre of M̄ 2-fold, and the same must necessarily be true for the singular fibres (of M̃ and M̄) as well. •

4.2. A remark on 2-fold covers of Seifert manifolds : Observe that if M̃ → M is a 2-fold covering of Seifert manifolds (a Seifert structure on M induces one on M̃) such that every fibre of M is twice covered by a single fibre of M̃ (that is, the covering involution acts on M̃ by rotation of the fibres through π radians), then a fibre γ̃ in M̃ of type (α, β̃) must cover a fibre γ in M of type (α, β) where β = 2β̃ + cα for some c ∈ Z [NR, 1.3]. Thus, for such a covering to exist, every fibre γ of M must have ODD multiplicity α. It is this fact that neccessitates the introduction of the intermediate 2^{t}-fold covering M̄ → M.

It follows from this remark that, since M̃ → M̄ is such a twofold covering, the singular fibres of M̄ have odd multiplicities. Also recall that, by Lemma 3.6, we may assume that each multiplicity of a singular fibre in M is either odd already or 2^{t}s_{i} for s_{i} odd. Then in the covering M̄ → M each regular fibre of M lifts to 2^{t} distinct fibres, as do the singular fibres of odd multiplicity, while those fibres of even multiplicity 2^{t}s_{i} must each be 2^{t}-fold covered by a single fibre of multiplicity s_{i}. Considering the base spaces F and F̄ as orbifolds, this covering induces a covering of orbifolds F̄ → F, or a 2^{t}-fold branched covering of surfaces with a branch set consisting of the 2b points corresponding to the 2b singular fibres with even multiplicity. Thus if F = P_{k}, then F̃ = F̄ is also a nonorientable surface Pk̃ with genus k̃ calculated as follows. As for the statement of Theorem 1.1, define b_{S} = 0 if every fibre has odd multiplicity (t = 0) and otherwise

b_{S} = b − 1 = \frac{(# of fibres in M of EVEN multiplicity) − 2}{2}.

Then, when t ≠ 0, χ(F̄) = 2^{t}(χ(F) − 2b_{S} − 2) + 2b_{S} + 2 which gives that k̃ = 2^{t}(k + 2b_{S}) − 2b_{S} (since χ(P_{k}) = 2 − k). Note however that this formula also holds when t = 0 in which case k̃ = k. Finally, if ε, ε̄, ε̃ are the generalised Euler invariants for M, M̄, M̃ respectively, then ε̃ = \frac{ε̄}{2} = \frac{2^{t}ε}{2}, as in [NR, 1.2] or [JN]. Now, with the number b_{S} defined above, we have the following lemma. (Here β_{i}(X) = dimH_{i}(X; Q) = dimH^{i}(X; Q) is the i^{th} Betti number.)

4.3. Lemma : In Lemma 4.1, we have β_{1}(X̃) ≤ 2^{t+1}(β_{1}(X) + b_{S}).

Proof : More generally consider a p^{s}-fold covering X̃ → X for p prime, where, putting K_{0} = π_{1}(X̃) and G_{0} = π_{1}(X), we have an exact sequence

1 ⟶ K_{0} ⟶ G_{0} ⟶^{α_{π}} Z/p^{s}Z ⟶ 1 .

We wish to estimate β_{1}(X̃) = rank K_{0}^{ab} in terms of β_{1}(X) = rank G_{0}^{ab}. Now τ = {x ∈ K_{0} : x^{n} ∈ [K_{0}, K_{0}] , for some n ∈ Z} defines a subgroup of K_{0} which is normal in G_{0}. (For x ∈ τ, g ∈ G_{0}, we have (gxg^{−1})^{n} = gx^{n}g^{−1} ∈ [K_{0}, K_{0}] for some n, since x^{n} ∈ [K_{0}, K_{0}] and [K_{0}, K_{0}] ◁ G_{0}, which follows from K_{0} ◁ G_{0}). Thus, writing K = K_{0}/τ = Z^{r}, where r = β_{1}(X̃), and G = G_{0}/τ we have an exact sequence 1 ⟶ K ⟶ G ⟶ Z/p^{s}Z ⟶ 1. Here G^{ab} is a quotient of G_{0}^{ab} = H_{1}(X), since G is a quotient of G_{0}, and in fact we may check that G^{ab} and G_{0}^{ab} have the same rank. Now K ◁ G implies that pK ◁ G, and we may factor out by pK to get an exact sequence

1 ⟶ K̄ ⟶ Ḡ ⟶^{ᾱ_{π}} Z/p^{s}Z ⟶ 1.  (✠)

Here K̄ = (Z/pZ)^{r}, and Ḡ ≅ ⟨ζ, K̄ | ζ^{p^{s}} ∈ K̄, ζ^{−1}xζ = T x, for x ∈ K̄⟩ , where ζ acts on K̄ = F_{p}^{r} (considered now as an F_{p}-vector space of dimension r) via some T ∈ GL(r, F_{p}). Let Γ = F_{p}[T] be the polynomial ring, which acts on K̄ in the obvious way, so that we may also view K̄ as a Γ-module. Now, since ζ^{p^{s}} ∈ K̄ and K̄ is abelian, T^{p^{s}}x = ζ^{−p^{s}}xζ^{p^{s}} = x. That is K̄ is annihilated by T^{p^{s}} − 1 = (T − 1)^{p^{s}} ∈ Γ (this equality holds in Γ since the coefficients are mod p). Since Γ is a P.I.D, we have K̄ = ⨁_{i=1}^{p^{s}}(Γ/(T − 1)^{i})^{e_{i}}, for some e_{i} ≥ 0. Each factor Γ/(T − 1)^{i} ≅ F_{p}^{i} as an F_{p}-vector space, so that β_{1}(X̃) = dimK̄ = Σ_{i=1}^{p^{s}}ie_{i} ≤ p^{s}Σe_{i}. On the other hand, Ḡ^{ab} ≅ ⟨ζ, K̄/(T − 1)K̄ | ζ^{p^{s}} ∈ K̄/(T − 1)K̄⟩, and K̄/(T − 1)K̄ = ⨁_{i=1}^{p^{s}}(Γ/(T − 1))^{e_{i}} ≅ F_{p}^{R} where R = Σe_{i}. Thus, so far, β_{1}(X̃) ≤ p^{s}R, and we have a surjective homomorphism ξ : G_{0}^{ab} → G^{ab} → Ḡ^{ab}, where Ḡ^{ab} = ⟨ζ, (Z/pZ)^{R} | ζ^{p^{s}} ∈ (Z/pZ)^{R}⟩.

In our case of interest p = 2 and s = t + 1, and we take α , α_{π} as in the proof of Lemma 4.1. Also, since τ(M) is a direct double τ(X) ⊕ τ(Y) [KK, 6.1], we may assume that τ(X) has exactly b_{S} + 1 factors of even order. Furthermore, there exists some element of τ(X) of order 2^{t+1}, q say, whose image α(q) generates Z/2^{t+1}Z. By the construction of Ḡ we have a surjection ξ̂ which induces ξ in the following commuting diagram,

[figure: G_{0}; \overline{ab}; G_{0}^{ab} = H_{1}(X); α]
[figure: ξ̂; ξ]
[figure: Ḡ; \overline{ab}; Ḡ^{ab}; \overline{ᾱ}; Z/2^{t+1}Z]

where ᾱ is induced by ᾱ_{π} of the exact sequence (✠), and ker(ᾱ) = K̄. Clearly we can choose q such that α(q) = ᾱ(ζ), still of order 2^{t+1}. Then ξ(q) = ζ + κ for some κ ∈ K̄, and ξ(2^{t+1}q) = 2^{t+1}ζ + 2^{t+1}κ = 2^{t+1}ζ since κ has order 2. But this must be 0 since q has order 2^{t+1}, and consequently

Ḡ^{ab} = ⟨ζ | 2^{t+1}ζ = 0⟩ ⊕ (Z/2Z)^{R}.

Then ξ taken “mod 2” gives a surjection (Z/2Z)^{β_{1}(X)+b_{S}+1} → (Z/2Z)^{R+1}, and hence R ≤ β_{1}(X) + b_{S}. But then β_{1}(X̃) ≤ 2^{t+1}(β_{1}(X) + b_{S}). •

4.4. Definition : By a G-manifold (G, Σ) we shall mean an oriented manifold Σ together with a finite group G which acts on Σ by orientation preserving homeomorphisms.

In this context we shall only consider Σ to be a connected 4-manifold, and suppose G = Z/2Z acts by an involution g : Σ → Σ. Furthermore we shall aim to construct Z/2Z-manifolds where the fixed point set of g is a closed connected surface F, smoothly embedded in Σ, such that g acts smoothly on a neighbourhood of F. That is F will have a smooth D^{2}-bundle neighbourhood N(F) in Σ, on which g acts by rotation of the fibres. We shall call such a surface a “good” fixed point set. Given such F, denote by e(F) the Euler class of the normal bundle to F in Σ, which is just the Euler class of ∂N(F) considered as a circle bundle over F. Now the Z/2Z-signature theorem states that, for a closed Z/2Z-manifold Σ with “good” fixed point set F, we have

sign(g, Σ) = e(F) .

Here sign(g, Σ) denotes the signature of the bilinear form ( ,)_{g} on H^{2}(Σ; Q) given by (σ , τ)_{g} = σ ∪ g^{∗}(τ). We may look to various sources [JO], [G] for a statement of this theorem for Σ smooth, and g a diffeomorphism. However the smoothness is required (by the proof in [JO] for example) only in the neighbourhood of the fixed point set. Thus the signature theorem holds equally well in the present setting.

In proving Theorem 1.1 we shall also use the fact that |sign(g, Σ)| ≤ β_{2}(Σ), and equals β_{2}(Σ) mod 2, for Σ a closed 4-manifold.

Observe, for example, that the covering spaces X̃ constructed in Lemma 4.1 are Z/2Z-manifolds, where the action is by the covering involution ψ_{X} which rotates the fibres in the boundary by π radians. Any D^{2}-bundle over a surface F may be considered as a Z/pZ-manifold for any prime p. In this case, the action is by rotation of the disk fibre, and F becomes the “good” fixed point set. Consequently, if M̃ in Lemma 4.1 is a circle bundle with Euler number ẽ, then the associated disk-bundle DM̃ is a Z/2Z-manifold with boundary (Z/2Z, M̃).

Thus we may construct a closed Z/2Z-manifold ΣM̃ = X̃ ∪M̃ DM̃, which has a “good” fixed point set F with e(F) = ẽ, and we have sign(g, ΣM̃) = ẽ. Note that this is the case M an S^{1}-bundle with Euler class e = 2ẽ.

More generally, if M̃ is Seifert fibred then the associated ‘disk’-bundle, or more precisely the mapping cylinder

CM̃ = M̃ × [0, 1]/((x_{1}, 0) ∼ (x_{2}, 0) for x_{1}, x_{2} in the same fibre),

is not in general a manifold. In particular, it ceases to be a manifold precisely at the isolated points p_{i} = (x_{i}, 0) for x_{i} in the singular fibre γ_{i} of multiplicity > 1. However, we may modify CM̃ by removing a neighbourhood which contains all such points. In this way we attempt to generalise the construction of the previous paragraph.

As in the statement of the theorem, given S = {γ_{i}(α_{i}, β_{i}) : i = 1, .., n}, let M_{1} = M_{S}(k_{1}; ε_{1}) be fibred over F_{1}, and M_{2} = M_{S}(k_{2}; ε_{2}) over F_{2}, and suppose that both M_{1} and M_{2} embed in S^{4}. Then, we may write S^{4} = X_{1} ∪_{M_{1}} Y_{1}, where we may suppose there is a covering ψ_{X_{1}} : X̃_{1} → X_{1} as described in Lemma 4.1 (and similarly for M_{2}).

Consider CM̃_{1}. For each i = 1, ..., n, let x_{i} ∈ F̃_{1} correspond to the fibre γ̃_{i} of type (s_{i}, β̃_{i}) in M̃_{1}. That is x_{i} = π̃(γ̃_{i}) where π̃ : M̃_{1} → F̃_{1} is the fibre map. There exists a disk δ ∈ F̃_{1} such that x_{i} ∈ δ for each i = 1, ..., n. Let B_{1} = π̃^{−1}(δ) × [0, 1/2]/(∼) ⊂ CM̃_{1}. This defines a neighbourhood which meets F̃_{1} in δ (note that CM̃_{1} contains F̃_{1} as {(x, 0) ∈ CM̃_{1}}). Also B_{1} is contractible to δ ∼ ∗, so that χ(B_{1}) = 1.

4.5. Claim : The boundary ∂B_{1} is a rational homology sphere.

(H_{k}(∂B_{1}; Q) = Q, 0, 0, Q)

Proof: We may write ∂B_{1} as a G.S. fibred manifold [JN]

∂B_{1} ≅ M(S^{2}; (0, 1), (s_{1}, β̃_{1}), (s_{2}, β̃_{2}), …, (s_{n}, β̃_{n})),

with a fundamental group which abelianises to a group of finite order :

\begin{aligned}π_{1}(∂B_{1}) & ≅ ⟨q_{0}, q_{1}, .., q_{n}, h | [q_{i}, h] = 1, q_{i}^{s_{i}}hβ̃_{i} = 1, h = 1, q_{0} = Π_{i=1}^{n}q_{i}⟩ \\ & ≅ ⟨q_{1}, .., q_{n} | q_{i}^{s_{i}} = 1⟩ . \end{aligned}

Thus H_{1}(∂B_{1}; Q) = 0, and the rest follows by Poincaré duality, given that ∂B_{1} is closed, orientable and connected. Note that in fact ∂B_{1} is a connected sum of lens spaces. •

Define the ‘modified associated disk-bundle’ to be DM̃_{1} = CM̃_{1} \ int(B_{1}), and let Σ_{1} = X̃_{1} ∪M̃_{1} DM̃_{1} define a Z/2Z-manifold with boundary ∂B_{1}. Here g acts on Σ_{1} via the covering automorphism of X̃_{1}, from Lemma 4.1, which extends across DM̃_{1} in the obvious way, with fixed point set F̃_{1} \ int(δ). Similarly construct the manifold Σ_{2} with boundary ∂B_{2}. Note that M̃_{1} and M̃_{2} may be assumed, by a Seifert equivalence if necessary to have exactly the same collection of strictly singular fibre types. In fact, for i = 1, 2 , M̃_{i} = M(F̃_{i}; (s_{1}, 2β_{1}), .., (s_{n}, 2β_{n}), (1, e_{i})), and ε̃_{1} −ε̃_{2} = e_{1} −e_{2}. Thus B_{1} ≅ B_{2} (≅ B say) in the strongest possible sense (that is by a homeomorphism which preserves the fibres, the set F̃_{i} ∩ B_{i}, and the group action), and we may take the union

Σ = Σ_{1} ⋃_{∂B} −Σ_{2} .

to be a Z/2Z-manifold with “good” fixed point set F = F̃_{1}♯F̃_{2}. Furthermore, F has Euler class e(F) = ε̃_{1} − ε̃_{2} = \frac{2^{t}(ε_{1}−ε_{2})}{2}. Note that ∂(−Σ_{2}) ≅ −∂B, while ∂(Σ_{1}) ≅ ∂B. However, to consistently orient the union of oriented manifolds we require the attaching homeomorphism to be orientation reversing, and here it is taken to be the obvious one. Now, by the Z/2Z-signature theorem we may write

sign(g, Σ) = \frac{2^{t}(ε_{1} − ε_{2})}{2}.

This signature is bounded by the value of β_{2}(Σ), independently of the ε_{i}. It remains to express this constraint in terms of k_{1} and k_{2}, and the constants t and b_{S}, which depend only on the set S.

4.6. Claim : χ(Σ) = 2 − 2(β_{1}(Σ_{1}) + β_{1}(Σ_{2})) + β_{2}(Σ).

Proof: By the Mayer-Vietoris sequence (with coefficients in Q),

[figure: H_{1}(∂B); H_{1}(Σ_{1}) ⊕ H_{1}(Σ_{2}); \overline{≅}; H_{1}(Σ); \overline{0}; H_{0}(∂B)]
∥  ∥
0  Q

we have β_{1}(Σ) = β_{1}(Σ_{1})+β_{1}(Σ_{2}), and, since Σ is a closed orientable 4-manifold, we may apply Poincaré duality to give

χ(Σ) = 2 − 2β_{1}(Σ) + β_{2}(Σ)
= 2 − 2(β_{1}(Σ_{1}) + β_{1}(Σ_{2})) + β_{2}(Σ) .  •

4.7. Claim : χ(Σ) = 2^{t+1}χ(X_{1}) + 2^{t+1}χ(X_{2}) + χ(F̃_{1}) + χ(F̃_{2}) − 2.

Proof: Since Σ is the union X̃_{1} ∪ DM̃_{1} ∪ DM̃_{2} ∪ X̃_{2} along closed 3-manifolds,

χ(Σ) = χ(X̃_{1}) + χ(X̃_{2}) + χ(DM̃_{1}) + χ(DM̃_{2}) .

Also, for each i = 1, 2, the union CM̃_{i} = DM̃_{i} ∪ B_{i} gives χ(CM̃_{i}) = χ(DM̃_{i}) + χ(B_{i}). But that is χ(DM̃_{i}) = χ(F̃_{i}) − 1 , for each i = 1, 2 , since CM̃_{i} is contractible to F̃_{i}, and χ(B_{1}) = χ(B_{2}) = 1. Thus,

χ(Σ) = χ(X̃_{1}) + χ(X̃_{2}) + χ(F̃_{1}) + χ(F̃_{2}) − 2
= 2^{t+1}χ(X_{1}) + 2^{t+1}χ(X_{2}) + χ(F_{1}) + χ(F_{2}) − 2 .  •

Combining 4.6 and 4.7 we have

β_{2}(Σ) = 2(β_{1}(Σ_{1}) + β_{1}(Σ_{2})) + 2^{t+1}(χ(X_{1}) + χ(X_{2})) + χ(F̃_{1}) + χ(F̃_{2}) − 4
= Ξ_{1} + Ξ_{2} ,  where  Ξ_{i} = 2β_{i}(Σ_{i}) + 2^{t+1}χ(X_{i}) + χ(F̃_{i}) − 2 .

4.8. Claim : For i = 1, 2, β_{1}(Σ_{i}) ≤ β_{1}(X̃_{i}) ≤ 2^{t+1}(β_{1}(X_{i}) + b_{S}).

Proof: Write Σ_{i}^{0} = Σ_{i} ∪_{∂B_{i}} B_{i}, for i = 1, 2. That is Σ_{i}^{0} = X̃_{i} ∪M̃_{i} CM̃_{i}, where CM̃_{i} is contractible to F̃_{i}. Then we have the Mayer-Vietoris sequence

[figure: H_{1}(M̃_{i}); \overline{ϕ}; H_{1}(X̃_{i}) ⊕ H_{1}(F̃_{i}); H_{1}(Σ_{i}^{0}); 0]

where ϕ = (î_{∗}, ĵ_{∗}) is induced by the inclusions î: M̃_{i} ↪ X̃_{i} and ĵ : M̃_{i} ↪ C̃M̃_{i}. But ĵ : H_{1}(M̃_{i}) → H_{1}(F̃_{i}) is a surjection, so that dim(Imϕ) ≥ β_{1}(F_{i}). But then,

\begin{aligned}β_{1}(Σ_{i}^{0}) & = β_{1}(X̃_{i}) + β_{1}(F̃_{i}) − dim(Imϕ) \\ & ≤ β_{1}(X̃_{i}) . \end{aligned}

Thus, by Lemma 4.3, we have β_{1}(Σ_{i}^{0}) ≤ 2^{t+1}(β_{1}(X_{i}) + b_{S}).

On the other hand, the Mayer-Vietoris sequence for Σ_{i}^{0} = Σ_{i} ∪_{∂B_{i}} B_{i},
[figure: H_{1}(∂B_{i}); H_{1}(Σ_{i}); ⊕H_{1}(B_{i}); \overline{≅}; H_{1}(Σ_{i}^{0}); \overline{0}; H_{0}(∂B_{i})]
∥  ∥  ∥
0  0  Q

together with Claim 4.5 gives β_{1}(Σ_{i}) = β_{1}(Σ_{i}^{0}), and we are done. •

Now, whenever S^{4} = X ∪_{M} Y, the Mayer-Vietoris sequence (with coefficients in Z) gives H_{i}(M) ≅ H_{i}(X) ⊕ H_{i}(Y) if i = 1, 2, and otherwise

H_{i}(X) ≅ H_{i}(Y) ≅ \{ \begin{matrix}0, \\ Z, \end{matrix} \begin{matrix}if i ≥ 3 , \\ if i = 0 . \end{matrix}

Also, since M ↪ X consequently induces epimorphisms H_{∗}(M) → H_{∗}(X), the relative exact sequence for (X, M) gives β_{1}(M) = β_{1}(X) + β_{2}(X, M) which becomes β_{1}(M) = β_{1}(X) + β_{2}(X) by Lefschetz duality.

So here we have that β_{1}(X_{i}) + β_{2}(X_{i}) = β_{1}(M_{i}) = k_{i} − 1, and β_{3}(X_{i}) = β_{4}(X_{i}) = 0. With 4.8 (for i = 1, 2), and these relations we now have

Ξ_{i} ≤ 2^{t+2}(β_{1}(X_{i}) + b_{S}) + 2^{t+1}χ(X_{i}) + χ(F̃_{i}) − 2
= 2^{t+1}(1 + β_{1}(X_{i}) + β_{2}(X_{i}) + 2b_{S}) + χ(F̃_{i}) − 2
= 2^{t+1}(k_{i} + 2b_{S}) − k̃_{i} = K_{i} say.

Thus we have shown that β_{2}(Σ) ≤ K_{1} + K_{2} = K. Also Ξ_{i} ≡ χ(F̃_{i}) ≡ k̃_{i} ≡ K_{i} mod 2, so that

sign(g, Σ) ≡ β_{2}(Σ) ≡ K mod 2 ,
and  |sign(g, Σ)| ≤ β_{2}(Σ) ≤ K .

It now follows that sign(g, Σ) ∈ {K, K − 2, K − 4, …, −K + 2, −K}, and, since sign(g, Σ) = \frac{2^{t}(ε_{1}−ε_{2})}{2}, where ε_{1} − ε_{2} must be an integer, that

ε_{1} − ε_{2} ∈ \{\frac{2K}{2^{t}} , \frac{2K − 4}{2^{t}}, \frac{2K − 8}{2^{t}}, …, \frac{−2K + 4}{2^{t}}, \frac{−2K}{2^{t}} \} ∩ Z .

Finally note that, since k̃_{i} = 2^{t}(k_{i} +2b_{S})−2b_{S} (following Remark 4.2), we have K_{i} = 2^{t}(k_{i} + 2b_{S}) + 2b_{S}, and so K = 2^{t}(k_{1} + k_{2} + 4b_{S}) + 4b_{S}, which completes the proof. •

References

[FQ] M.H. Freedman, and F. Quinn, Topology of 4-Manifolds, Princeton University Press, Princeton (1990).

[GL] P. Gilmer and C. Livingston, ‘On embedding 3-manifolds in 4-space’, Topology 22 (1983), 241-252.

[G] C.McA. Gordon, ‘On the G-signature theorem in dimension four’, in A la Recherche de la Topologie Perdue, ed. Guillou and Marin, Birkhäuser, 1986.

[H] J.A. Hillman, ‘Embedding homology equivalent 3-manifolds in 4-space’, Research Report 93-22, University of Sydney (1993), (revised 1994).

[JN] M. Jankins and W.D. Neumann, Lectures on Seifert Manifolds, Brandeis Lecture Notes 2, (1983).

[JO] K. Jänich and E. Ossa, ‘On the signature of an involution’, Topology 8 (1969), 27-30.

[KK] A. Kawauchi and S. Kojima, ‘Algebraic classification of linking pairings on 3-manifolds’, Math. Ann. 253 (1980), 29-42.

[M] W.S. Massey, ‘Proof of a conjecture of Whitney’, Pacific J. Math. 31 (1969), 143-156.

[NR] W.D. Neumann and F. Raymond, ‘Seifert manifolds, plumbing, μ-invariant and orientation reversing maps’, in Proceedings Sympos., Santa Barbara, CA 1977, Lecture Notes in Math. 664, Springer Verlag (1978) 163-196.

[O] P. Orlik, Seifert Manifolds, Lecture Notes in Math. 291, Springer Verlag (1972).

[R1] D. Rolfsen, Knots and Links, Publish or Perish, Berkeley (1976).

[R2] D. Rolfsen, ‘Rational surgery calculus: Extension of Kirby’s theorem’, Pacific J. Math. 110 (1984), 377-386.

[S] H. Seifert, ’Topology of 3-dimensional fibred spaces’, Acta Math. 60 (1933), 147-288.

[W] C.T.C. Wall, ’Quadratic forms on finite groups, and related topics’, Topology 2 (1964), 281-298.

[Zee] E.C. Zeeman, ‘Twisting spun knots’, Trans. A.M.S. 115 (1965), 471-495.

To the memory of my father, C.F.C Crisp, Sydney, November 17th, 1994.