Embedding 3-manifolds with circle actions in 4-space

Jonathan A. Hillman School of Mathematics and Statistics, The University of Sydney Sydney, NSW 2006, AUSTRALIA e-mail: jonh@ maths.usyd.edu.au

ABSTRACT

We show that if M is an orientable 3-manifold which is Seifert fibred over an orientable base orbifold B and which embeds in ℝ^{4} then the generalized Euler invariant of the fibration is determined up to sign by B unless H_{1}(M; ℤ) is torsion free, in which case it can take at most one nonzero value (up to sign). In particular, no such manifold with 3 exceptional fibres and Euler invariant 0 embeds in ℝ^{4}.

AMS Subject classification (1985): Primary 57N10. Secondary 57N13. Key words and phrases: embedding. Euler invariant. linking pairing. Seifert bundle.

If a closed 3-manifold M embeds in ℝ^{4} (as a TOP locally flat submanifold) it must be orientable, and the linking pairing on the torsion subgroup T (M) of H_{1}(M; ℤ) must be hyperbolic [KK]. In particular, T (M) ≅ U ⊕ U for some finite abelian group U [Ha]. In §1 we shall observe that this imposes strong constraints on the generalized Euler invariants of orientable 3-manifolds which are Seifert fibred over orientable base orbifolds (i.e., admit effective S^{1}- actions) and which embed in ℝ^{4}. The present simple argument does not work when the base orbifold is nonorientable; in [CH] we use the Z/2Z-Index Theorem to constrain the Euler invariants in the latter case.

In §2 we use Seifert invariants to describe the linking pairing, and in §3 we

Typeset by AMS-TEX

consider the cases with at most 3 exceptional fibres. In particular, no such manifold with 3 exceptional fibres and Euler invariant 0 embeds in ℝ^{4}.

§1. The torsion subgroup

We shall assume henceforth that M = M(g; S) is an orientable 3-manifold which is the total space of a Seifert bundle over the orbifold B = T_{g}(α_{1}, … α_{r}) (where T_{g} is the orientable surface of genus g), with Seifert invariants S = {(α_{i}, β_{i}) | 1 ≤ i ≤ r}, where 1 < α_{i} and (α_{i}, β_{i}) = 1, for all i ≤ r. (We do not assume that 0 < β_{i} < α_{i}). If r = 1 we allow also the possibility α_{1} = 1. Let ε = −Σ_{i=1}^{i=r}(β_{i}/α_{i}) be the generalized Euler invariant of the Seifert bundle, and let Π = Π_{i=1}^{i=r}α_{i}. As B is the connected sum of T_{g} and S^{2}(α_{1}, … α_{r}) we have M = M(0; S)♯_{f}M(g; ∅) = M(0; S)♯_{f}(T_{g} × S^{1}), where ♯_{f} denotes fibre sum [CH].

Theorem. Let M = M(g; S) be an orientable 3-manifold which is Seifert fibred over an orientable base orbifold B = T_{g}(α_{1}, … α_{r}). Then H_{1}(M; ℤ) ≅ Z^{2g} ⊕ (⨁_{i≥0}(Z/λ_{i}Z)), where λ_{i} is determined by {α_{1}, … α_{r}} and is nonzero, for all i > 0, while |ε|Π = λ_{0}Π_{j≥1}λ_{j}. Proof. The fundamental group π_{1}(M) has a presentation ⟨a_{1}, … b_{g}, q_{1}, … q_{r}, h | (Π[a_{i}, b_{i}])(Πq_{j}) = 1, q_{i}^{α_{i}}h^{β_{i}} = 1, h central⟩. Hence H_{1}(M; ℤ) ≅ Z^{2g} ⊕ Cok(A), where A is the matrix

A = ( \begin{matrix}0 & 1 & … & 1 \\ β_{1} & α_{1} & … & 0 \\ ⋮ & 0 & ... & ⋮ \\ β_{r} & 0 & … & α_{r} \end{matrix} ) .

Let E_{i}(A) be the ideal generated by the (r+1−i)×(r+1−i) subdeterminants of A, and let Δ_{i} be the positive generator of E_{i}(A). Then Δ_{0} = |det(A)| = |ε|Π, while as the elements of each row are relatively prime Δ_{i} is the highest common factor of the (r − i − 1)-fold products of distinct α_{j}s, if 0 < i < r, and Δ_{i} = 1 if i ≥ max{r − 1, 1}. (In particular, if r > 2 then Δ_{r−2} is the highest common factor of {α_{1}, … α_{r}}). Thus Δ_{i} depends only on {α_{1}, … α_{r}} and is nonzero, for all i > 0. If we set λ_{i} = Δ_{i}/Δ_{i+1}, for i ≥ 0, then Cok(A) ≅ ⨁_{i≥0}(Z/λ_{i}Z), by the Elementary Divisor Theorem. In particular, |ε|Π = λ_{0}Π_{j≥1}λ_{j}. □

Note that T (M) ≅ T (M(0; S)) and h ∈ T (M) if and only if ε ≠ 0.

The elementary divisors λ_{i} may be determined more explicitly by localization. Let p be a prime, and let v_{i} ≥ 0 be the p-adic valuation of α_{i}. (Thus α_{i} = p^{v_{i}}q_{i}, where p does not divide q_{i}). If the indexing is such that v_{i} ≥ v_{i+1} for all i then the p-adic valuation of λ_{j} is v_{j+2}, for all j ≥ 1. In particular, λ_{i+1} divides λ_{i}, for all i ≥ 0.

Corollary. If T (M) is a nontrivial direct double then T (M) and |ε| are determined by {α_{1}, … α_{r}}. Proof. If ε = 0 then T (M) ≅ ⨁_{i≥1}(Z/λ_{i}Z). Hence if T (M) is a direct double λ_{2i−1} = λ_{2i}, for all i > 0. If ε ≠ 0 and T (M) is a direct double λ_{2i} = λ_{2i+1}, for all i ≥ 0, which is possible for at most one nonzero value of ε (up to sign). Clearly these two systems of equations can only be both satisfied if λ_{i} = 1 for all i > 0 and λ_{0} = 0 or 1, in which case T (M) = 0. □

When M is Seifert fibred over a nonorientable base orbifold T (M) is again largely determined by the set {α_{1}, … α_{r}}, but ε is not constrained at all by the condition that T (M) be a direct double (see [CH]).

Although the β_{i}s only contribute to the structure of T (M) via ε, they play a more substantial role in the linking pairing. (See below).

If T (M) = 0 and ε ≠ 0 then M(0; S) is a ℤ-homology sphere, and so embeds as a TOP locally flat submanifold of ℝ^{4} [FQ]. Hence M = M(0; S)♯_{f}(T_{g} ×S^{1}) embeds also. If T (M) = 0 and ε = 0 then H_{1}(M(0; S); ℤ) ≅ Z, and M(0; S) again embeds in ℝ^{4} if moreover it admits a degree 1 map to S^{2} × S^{1} which induces isomorphisms on homology with local coefficients [Hi]. However, this is not a necessary condition, and it is not known in general which (Seifert fibred) homology S^{2} × S^{1}s embed.

If M embeds in ℝ^{4} then so does M♯_{f}(T_{g} × S^{1}). Is there a converse? The analogous implication in the case of nonorientable base orbifolds is not reversible. (See [CH]).

§2. Bilinear pairings

A linking pairing on a finite abelian group N is a symmetric bilinear function ℓ : N ×N → ℚ/ℤ which is nonsingular in the sense that : n ↦ ℓ(−, n) defines an isomorphism from N to Hom(N, ℚ/ℤ). Such a pairing splits uniquely as the orthogonal sum (over primes p) of its restrictions to the p-primary subgroups of N. It is metabolic if there is a subgroup P < N such that P = P^{⊥} = {t ∈ N | ℓ_{M}(t, p) = 0 ∀p ∈ P }, and is hyperbolic if N is the direct sum of two such subgroups.

If M is a closed oriented 3-manifold Poincaré duality determines a linking pairing ℓ_{M} : T (M) × T (M) → ℚ/ℤ, which may be described as follows. Let w, z be 1-cycles representing elements of T (M) and suppose that mz = ∂C for some 2-chain C which is transverse to z and some nonzero m ∈ ℤ. Then ℓ_{M}(w, z) = (w.C)/m ∈ ℚ/ℤ. It follows easily from the Mayer-Vietoris theorem and duality that if M embeds in ℝ^{4} then ℓ_{M} is hyperbolic. (If X and Y are the closures of the components of ℝ^{4} − M and T_{X} and T_{Y} are the kernels of the induced homomorphisms from T (M) to H_{1}(X; ℤ) and H_{1}(Y; ℤ) (respectively) then T (M) ≅ T_{X} ⊕T_{Y} and the restriction of ℓ_{M} to each of these summands is trivial [KK]).

There are analogous pairings on covering spaces of M. In particular, if ϕ : π_{1}(M) → Z is an epimorphism the homology groups of the associated covering space M_{ϕ} are Λ-modules, where Λ = ℤ[Z] = ℤ[t, t^{−1}]. There is a Blanchfield pairing on the Λ-torsion submodule of H_{1}(M_{ϕ}; ℤ) with values in ℚ(t)/Λ which is nonsingular modulo the maximal finite submodule and hermitean with respect to the involution sending t to t^{−1}. If M embeds in ℝ^{4} then this Blanchfield pairing is again hyperbolic [Ka].

Assume now that M = M(g; S). Then T (M) is a subgroup of the group generated by h and q_{1}, … q_{r}, and the Seifert structure gives natural 2-chains relating these 1-cycles. For let N_{i} be a torus neighborhood of the i^{th} singular fibre, with meridian D_{i}, and let B_{o} be a section of the restriction of the Seifert fibration to M − ∪_{i≥1}N_{i}. Then ∂D_{i} = α_{i}q_{i} + β_{i}h and ∂B_{o} = Σq_{i}. Moreover h.B_{o} = 1, h.D_{i} = 0 = q_{j}.B_{o}, q_{i}.D_{i} = β_{i} and q_{i}.D_{j} = 0 if i ≠ j, for all 1 ≤ i, j, ≤ r. As these intersection numbers are independent of g, it is clear that ℓ_{M} ≅ ℓ_{M(0;S)}.

For small values of r we can compute ℓ_{M} directly (see below). Assume that r ≥ 3. If ε ≠ 0 then h ∈ T (M) and M(0; S) is a ℚ-homology 3-sphere, and is a ℕil^{3}- or 𝕊̃𝕃-manifold. Let Δ = εΠ and δ = h.c.f.{α_{1}, … α_{r}}. Then Δh = ∂(ΠB − Σ(Π/α_{i})D_{i}) and Δα_{i}q_{i} = ∂(ΔD_{i} + β_{i}(Σ(Π/α_{j})D_{j}) − β_{i}ΠB). Hence ℓ_{M}(h, h) = Π/Δ, ℓ_{M}(h, q_{i}) = −β_{i}Π/Δα_{i}, ℓ_{M}(q_{i}, q_{i}) = β_{i}(α_{i}Δ + β_{i}Π)/Δ(α_{i})^{2} and ℓ_{M}(q_{i}, q_{j}) = β_{i}β_{j}Π/Δα_{i}α_{j}, if i ≠ j. Since T (M) is generated by h and the q_{i}s and ℓ_{M} is nonsingular h = 0 if and only if Δ divides Π and each of β_{i}Π/α_{i}, i.e., Δ = h.c.f.{Π/α_{1}, … Π/α_{r}}. In particular, H_{1}(M; ℤ) ≅ (Z/δZ)^{r−1} if and only if Δ = δ^{r−1}. In this case T (M) = H_{1}(M; ℤ) is generated by any r − 1 elements of {q_{1}, … q_{r}}, and h = 0. In general it may not be easy to find a minimal generating set for T (M), and we should first localize at a prime.

If r ≥ 3 and ε = 0 then M is a flat 3-manifold or a ℍ^{2}×𝔼^{1}-manifold, and h ∉ T (M), so M is the mapping torus F ×_{θ} S^{1} of a periodic self homeomorphism θ of a closed orientable surface F. (If g = 0 the bundle projection to S^{1} is essentially unique). Let ϕ : π_{1}(M) → S^{1} be the epimorphism induced by the bundle projection. Then M_{ϕ} ≅ F × R, and so H_{1}(M_{ϕ}; ℤ) ≅ H_{1}(F; ℤ) as an abelian group. Hence it is a torsion Λ-module (with no nontrivial finite submodule). The Blanchfield pairing is determined by the intersection form I_{F} on H_{1}(F; ℤ) and the isometry θ_{∗} = H_{1}(θ) [Tr]. In particular, it is hyperbolic if and only if H_{1}(F; ℤ) = A ⊕ B where θ_{∗}(A) = A, θ_{∗}(B) = B and I_{F} (A, A) = I_{F} (B, B) = 0. (Note that if M(0; S) = F ×_{θ} S^{1} then M(g; S) = (F ♯T_{g}) ×_{θ♯id} S^{1}, and so we may again concentrate on the case g = 0).

§3. The cases with r ≤ 3

If r = 1 then ε ≠ 0, H_{1}(M; ℤ) = T (M) = (Z/εαZ)h and ℓ_{M}(h, h) = −α/β. If r = 2 let x, y be such that xα_{1} − yβ_{1} = 1, and let k = yq_{1} + xh and k′ = α_{1}q_{1} + β_{1}h = ∂D_{1}. Then q_{1} = −β_{1}k + xk′ and h = α_{1}k − yk′, and εα_{1}α_{2}k = ∂(α_{2}B − (xα_{2} + yβ_{2})D_{1} − D_{2}). Hence H_{1}(M; ℤ) = (Z/εα_{1}α_{2}Z)k. If ε ≠ 0 then T (M) = H_{1}(M; ℤ) and ℓ_{M}(k, k) = (xα_{2} + yβ_{2})/εα_{1}α_{2}, while if ε = 0 then T (M) = 0. Thus if r ≤ 2 and T (M) is a direct double εΠ = 0 or ±1, and so M(0; S) ≅ S^{3} or S^{2} × S^{1}. Since these embed smoothly in ℝ^{4} so does M = M(0; S)♯_{f}(T_{g} × S^{1}) [CH].

The simplest nontrivial case is when r = 3. If r = 3 then Δ_{0} = |ε|Π, Δ_{1} = δ = h.c.f.(α_{1}, α_{2}, α_{3}) and Δ_{j} = 1 for j ≥ 2. We shall assume that g = 0 and T (M) is a direct double.

If ε = 0 then T (M) ≅ Z/δZ, and so δ = 1. The equation εΠ = 0 implies that α_{1} divides β_{1}α_{2}α_{3} (etc.), and hence that α_{1} = m_{2}m_{3}, α_{2} = m_{1}m_{3} and α_{3} = m_{1}m_{2}, where m_{i} = h.c.f.(α_{j}, α_{k}) (for any permutation {i, j, k} of {1, 2, 3}). Let μ = m_{1}m_{2}m_{3}. We have H_{1}(M; ℤ) ≅ Z, and the abelianization homomorphism ϕ : π = π_{1}(M) → Z is given by ϕ(h) = −μ, and ϕ(q_{i}) = m_{i}β_{i}, for 1 ≤ i ≤ 3. In this case M ≅ F ×_{θ} S^{1}, where χ(F) = m_{1} +m_{2} +m_{3} −μ and θ has order μ. (Note that χ(F) < 0 unless {m_{1}, m_{2}, m_{3}} = {1, 2, 3}, in which case M = M(0; (2, 1), (3, −1), (6, −1)) is the flat 3-manifold with holonomy Z/6Z).

The fundamental group π_{1}(M) has a presentation ⟨q_{1}, q_{2}, q_{3}, h | q_{1}q_{2}q_{3} = 1, q_{i}^{α_{i}}h^{β_{i}} = 1, h central⟩. Since h.c.f.(μ, m_{1}β_{1}) = m_{1} and h.c.f.(m_{1}, m_{2}β_{2}) = 1 there are integers e, f_{1} and f_{2} such that eμ + f_{1}m_{1}β_{1} + f_{2}m_{2}β_{2} = 1. Let t = h^{−e}q_{1}^{f_{1}}q_{2}^{f_{2}}, r_{i} = q_{i}t^{−m_{i}β_{i}} and s = ht^{μ}. Then ϕ(r_{i}) = ϕ(s) = 0 and ϕ(t) = 1, and π has an equivalent presentation ⟨r_{1}, r_{2}, r_{3}, s, t | t^{eμ−1}(r_{1}t^{m_{1}β_{1}})^{f_{1}}(r_{2}t^{m_{2}β_{2}})^{f_{2}} = s^{e}, (r_{i}t^{m_{i}β_{i}})^{α_{i}}(st^{−μ})^{β_{i}} = 1,

r_{1}t^{m_{1}β_{1}}r_{2}t^{m_{2}β_{2}}r_{3}t^{m_{3}β_{3}} = 1, t^{μ}r_{i}s = sr_{i}t^{μ}, st = ts⟩. We may use the free differential calculus to obtain a presentation matrix for the Λ-module H_{1}(M_{ϕ}; ℤ), with 4 generators and 9 relations. Although this can be simplified by Tietze moves to one with 2 generators and 6 relations, the structure of the module is not clear. However if we tensor over Λ with the field ℚ[ζ_{μ}] = ℚ[t, t^{−1}]/(ϕ_{μ}(t)) of μ^{th} roots of unity (via the homomorphism sending t ∈ Λ to ζ_{μ}) we find that the resulting ℚ[ζ_{μ}]-vector space has dimension 1, and so is not a direct double. Hence the Blanchfield pairing cannot be hyperbolic and no such manifold can embed in ℝ^{4}.

The situation is less clear when ε ≠ 0. We then have εΠ = ηδ^{2}, where η = ±1. On writing γ_{i} = α_{i}/δ we obtain the equation γ_{1}γ_{2}β_{3} + γ_{1}β_{2}γ_{3} + β_{1}γ_{2}γ_{3} = −η. Hence the γ_{i}s must be pairwise coprime. (That is, δ is also the h.c.f. of any two of the α_{i}s). Since T (M) ≅ (Z/δZ)^{2}, it is easily seen that ℓ_{M} is metabolic if and only if (δ)^{2}(ℓ_{M}(q_{1}, q_{2})^{2} − ℓ_{M}(q_{1}, q_{1})ℓ_{M}(q_{2}, q_{2})) = ηβ_{1}β_{2}β_{3} is a square modulo (δ). If moreover δ is odd metabolic forms on (Z/δZ)^{2} are hyperbolic, so ℓ_{M} is hyperbolic if and only if ηβ_{1}β_{2}β_{3} is a square modulo (δ).

If ε ≠ 0 and δ = 1 then M = M(0; (γ_{1}, β_{1}), (γ_{2}, β_{2}), (γ_{3}, β_{3})) is an homology sphere, and so always embeds in ℝ^{4}. The question of which such homology spheres embed smoothly is still open. (Compare Problem 4.2 of [Ki]). In particular, the Poincaré homology 3-sphere M(0; (2, 1), (3, −1), (5, −1)) does not embed smoothly. On the other hand, the rational homology spheres M(0; (2, 1), (2, 1), (2, −1)) and M(0; (3, 1), (3, 1), (3, −1)) do embed smoothly. These three manifolds are of particular interest for the strategy of [Hi], as their fundamental groups are virtually solvable. (The only other 3-manifolds which are Seifert fibred over S^{2}, have virtually solvable fundamental group and embed in ℝ^{4} are S^{3}, S^{2}×S^{1} and M(0; (2, 1), (2, 1), (2, −1), (2, −1)) [CH]).

References

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

[Hi] Hillman, J.A. Embedding homology equivalent 3-manifolds in 4-space, Math. Z. 223 (1996), 473-481.

[Ha] Hantzsche, W. Einlagerung von Mannigfaltigkeiten in euklidische Räume, Math. Z. 43 (1937), 38-58.

[Ka] Kawauchi, A. On quadratic forms of 3-manifolds, Invent. Math. 43 (1977), 177-198.

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

[Ki] Kirby, R.C. Problems in low-dimensional topology, in Geometric Topology (ed. W.H.Kazez), Part 2 (1997), 35-473.

[Tr] Trotter, H.F. On S-equivalence of Seifert matrices, Invent. Math. 20 (1973), 173-207.