VIII.5 The level structure of almost contact structures in dimension 5 163
Let θbe the group automorphism of H2(X;Z) given byθ(ei) = e0i for alli ≥ 1, and which is the identity on TorH2(X;Z). Then
(c0◦θ)(ei) =c0(e0i) =c(ei) ∀i≥1.
Hencec0◦θ=conG. This equality holds on all ofH2(X;Z)sincecandc0are homomorphism toZand hence vanish on all torsion elements. By the assumption above, this implies thatw2(X)◦θ=w2(X).
Moreover, sinceθis the identity on TorH2(X;Z), it preserves the linking form. By Barden’s theorem 7.16, the automorphism θis induced by an orientation preserving self-diffeomorphismφ:X −→ X such thatφ∗=θ. We have
c(λ) =c0(φ∗λ) = (φ∗c0)(λ), for allλ∈H2(X;Z).
Henceφ∗c0 =c.
We can use Theorem 8.17 or 8.18 (cf. also Lemma 8.8 and Definition 8.10) to get the following corollary for almost contact structures.
Corollary 8.22. LetX be a simply-connected, closed, oriented 5-manifold. Then two almost contact structuresξ0 andξ1 onX are equivalent if and only ifc1(ξ0)andc1(ξ1)have the same divisibility in integral cohomology.
The other direction follows, because the divisibilities of elements inH2(X;Z)are preserved under automorphisms. Note that simply-connected manifolds have torsion freeH2 by the Universal Coeffi-cient Theorem.
Definition 8.23. We denote the divisibility ofc1(ξ)byd(ξ), as in Definition 6.5.
We sometimes calld(ξ) the level of ξ. By Corollary 8.22, almost contact structures and hence contact structures on a simply-connected 5-manifoldXnaturally form a “spectrum” consisting of levels which are indexed by the divisibility of the first Chern class. Two contact structures onXare equivalent as almost contact structures if and only if they lie on the same level. Note that simply-connected spin 5-manifolds have only even levels and non-spin 5-5-manifolds only odd levels, cf. Lemma 8.8. In Chapter X, we will use invariants from contact homology to investigate the “fine-structure” of each level in this spectrum. For instance, O. van Koert [74] has shown that for many simply-connected 5-manifolds the lowest level, given by divisibility0, contains infinitely many inequivalent contact structures.
Chapter IX
Circle bundles over symplectic manifolds
Contents
IX.1 Topology of circle bundles . . . . 165 IX.2 Connections on circle bundles with prescribed curvature . . . . 170 IX.3 The Boothby-Wang construction . . . . 171
In the first part of this chapter, we collect and prove some results on the topology of circle bundles over closed manifoldsM. The results will be used in the case where the dimension of the base manifold is equal to4in Chapter X. In particular, we will show that the total space of a circle bundle is simply-connected if and only if the base manifold is simply-simply-connected and the Euler class is indivisible. We also determine when the total space is spin. If M is a simply-connected 4-manifold and the Euler class of the circle bundle overM is indivisible, we can use the classification of simply-connected 5-manifolds from Chapter VII to determine the total spaceXup to diffeomorphism. It turns out thatX is diffeomorphic to a connected sum of several copies ofS2×S3 ifXis spin. IfXis not spin there is an additional summand of the formS2טS3.
The second part of this chapter describes the so-called Boothby-Wang construction: Suppose that ω is a symplectic form on a manifold M which represents an integral cohomology class and let X be the total space of the circle bundle overM with Euler class equal to[ω]. The construction then associates to ω a contact structure on X. We will consider this construction in the Chapter X for symplectic 4-manifolds. By the classification of the total spacesXof circle bundles mentioned above, one can choose many different simply-connected symplectic4-manifoldsMwhich give diffeomorphic simply-connected5-manifoldsXand hence many contact structures on the same abstract5-manifold.
We will show that in some cases this gives rise to contact structures on simply-connected 5-manifolds, coming from different symplectic4-manifolds, which are equivalent as almost contact structures but not equivalent as contact structures .
IX.1 Topology of circle bundles
LetM be a closed, connected, orientedn-manifold andπ:X → M the total space of a circle bundle overMwith Euler classe∈H2(M;Z), whereH2(M;Z)might have torsion. We consider the map
he,−i:H2(M;Z)→Z,
given by evaluation of the Euler class. We make the following generalization of definition 6.5 for this case:
Definition 9.1. We calleindivisible ifhe,−iis surjective.
Clearly, ifeis indivisible,ecannot be written ase=kc, withk >1andc∈H2(M;Z).
Lemma 9.2. A classe∈H2(M;Z)is indivisible if and only if the map e∪:Hn−2(M;Z)→Hn(M;Z)∼=Z is surjective.
Proof. The mape∪onHn−2(M;Z)is surjective if and only if there exists an elementα∈Hn−2(M;Z) such that
he∪α,[M]i= 1.
Via Poincar´e duality (c := α∩[M]) this is equivalent to the existence of a classc ∈H2(M;Z)such that
he, ci= 1, which is equivalent to the maphe,−ibeing surjective.
There is the following exact Gysin sequence for circle bundles [100]:
. . .−→π∗ Hk(X)−→π∗ Hk−1(M)−→∪e Hk+1(M)−→π∗ Hk+1(X)−→π∗ . . .
Lemma 9.3. Integration along the fibre π∗: Hk+1(X) → Hk(M) is Poincar´e dual to the map π∗:Hn−k(X)→Hn−k(M).
Proof. Letπ:D→Mdenote the disc bundle with Euler classe. ThenX∼=∂Dand integration along the fibre
π∗:Hk+1(∂D)→Hk(M) is given by (see [100])
Hk+1(∂D)−→δ Hk+2(D, ∂D)−→τ−1 Hk(D)π
∼∗
=Hk(M).
Hereδdenotes the connecting homomorphism in the long exact sequence of the pair(D, ∂D)andτ−1 the inverse of the Thom isomorphism
τ:Hk(D)→Hk+2(D, ∂D), x7→x∪u,
where the Thom classu∈H2(D, ∂D)can be written as the Poincar´e dual of the fundamental class of the zero sectionN inD. Under Poincar´e duality, the connecting homomorphismδcorresponds to
i∗:Hn−k(∂D)→Hn−k(D),
wherei:∂D→Dis the inclusion. We want to show thatπ∗i∗:Hn−k(∂D)→Hn−k(M)is Poincar´e dual to integration along the fibre. This is equivalent to
π∗◦P D◦τ ◦π∗:Hk(M)→Hn−k(M),
IX.1 Topology of circle bundles 167 whereP D:Hk+2(D, ∂D) → Hn−k(D) is Poincar´e duality, being just Poincar´e duality onM. Let α∈Hk(M). Then
π∗◦P D◦τ◦π∗(α) =π∗((π∗α∪u)∩[D])
=π∗(π∗α∩(u∩[D]))
=π∗(π∗α∩[N]) =α∩π∗[N]
=α∩[M].
This proves the claim.
Lemma 9.4. The image ofπ∗:H2(X;Z)→H2(M;Z)is the kernel ofhe,−i. Proof. We consider the following part of the Gysin sequence:
Hn−1(X)−→π∗ Hn−2(M)−→∪e Hn(M)∼=Z.
A classα ∈Hn−2(M;Z)is in the image ofπ∗ if and only ife∪α= 0, which is the case if and only if the Poincar´e dualc=P D(α)∈H2(M;Z)satisfieshe, ci= 0. Since integration along the fibre
π∗:Hn−1(X;Z)→Hn−2(M;Z) is by Lemma 9.3 Poincar´e dual to
π∗:H2(X;Z)→H2(M;Z), this proves the claim.
We now consider the following part of the Gysin sequence:
. . .−→Hn−2(M)−→∪e Hn(M)−→Hn(X)−→π∗ Hn−1(M)−→0.
This shows thateis indivisible if and only ifπ∗: Hn(X;Z) → Hn−1(M;Z)is an isomorphism, in other words
π∗:H1(X;Z)−→H1(M;Z)
is an isomorphism. The long exact homotopy sequence of the fibrationS1 →X →M . . .−→π2(M)−→∂ π1(S1)−→π1(X)−→π∗ π1(M)−→1 induces via Lemma A.5 an exact sequence
H1(S1;Z)−→H1(X;Z)−→H1(M;Z)−→0.
Hence we see thateis indivisible if and only if the fibreS1 ⊂Xis null-homologous.
From the long exact homotopy sequence above, we see that the fibre is null-homotopic if and only if∂:π2(M)→π1(S1)is surjective. Both statements are equivalent to
π∗:π1(X)→π1(M) being an isomorphism.
Lemma 9.5. The map∂:π2(M)→π1(S1)∼=Zin the long exact homotopy sequence for fibre bundles is given by
π2(M)−→h H2(M;Z)he,−i−→ Z wherehdenotes the Hurewicz homomorphism.
Proof. Let f: S2 → M be a continous map and E = f∗X the pull-back S1-bundle over S2. By naturality of the long exact homotopy sequence there is a commutative diagram
π2(X) −−−−→ π2(M) −−−−→∂ π1(S1) −−−−→ π1(X) −−−−→ π1(M) x
f∗ x
=
x
x
x
π2(E) −−−−→ π2(S2) −−−−→∂ π1(S1) −−−−→ π1(E) −−−−→ 1
Since f can represent any element inπ2(M)and the equationf∗(e(X)) = e(E) holds by naturality of the Euler class it suffices to prove the claim forM equal toS2. We then have to prove that the map
∂:π2(S2)→π1(S1)is multiplicationZ→a· Zby the Euler numbera=he(E),[S2]i.
By the exact sequence above it follows thatπ1(S1) = Zmaps surjectively ontoπ1(E). Hence π1(E) is a finite cyclic group, in particular abelian. Therefore we have to prove that H2(E) ∼= H1(E)∼=π1(E)is equal toZ/aZ. This follows from the following part of the Gysin sequence:
H0(S2)−→∪e H2(S2)−→H2(E)−→π∗ H1(S2) = 0.
Lemma 9.5 implies that∂is surjective if and only ifhe,−iis surjective on spherical classes.
Remark 9.6. More generally, letX→M be aU(m)-principal bundle. Using the clutching construc-tion and a Mayer-Vietoris argument one can show that π1(E) = H1(E) = Z/aZfor any principal bundleU(m)→E →S2, wherea=hc1(E),[S2]i. This implies as above (this lemma has been used in the proof of Theorem 8.18):
Lemma 9.7. LetX →M be aU(m)-principal bundle. Then the map∂ :π2(M) →π1(U(m))∼=Z in the long exact homotopy sequence is given by
π2(M)−→h H2(M;Z)hc1−→(X),−iZ.
Lemma 9.8. Xis simply-connected if and only ifM is simply-connected andeis indivisible.
Proof. If X is simply-connected, the long exact homotopy sequence shows that π1(M) = 1 and
∂:π2(M)→π1(S1)is surjective. HenceM is simply-connected and the Hurewicz maph:π2(M)→ H2(M;Z)is an isomorphism. The surjectivity of∂ implies thateis indivisible. Conversely, suppose thatMis simply-connected andeis indivisible. The same argument shows that∂is surjective. The long exact homotopy sequence then implies the exact sequence1→π1(X)→1. Henceπ1(X) = 1.
Lemma 9.9. Suppose the first Betti number ofMvanishes,b1(M) = 0. Then the mapπ∗:H2(M;Z)→ H2(X;Z)is surjective with kernelZ·e.
Proof. We consider the following part of the Gysin sequence:
H0(M)−→∪e H2(M)−→π∗ H2(X)−→H1(M).
By assumption,H1(M) = 0. Henceπ∗:H2(M) → H2(X)is surjective with kernelH0(M)∪e= Z·e.
IX.1 Topology of circle bundles 169 We now determine when the total spaceXis spin.
Lemma 9.10. The total spaceXis spin if and only ifw2(M) ≡αemod2for someα∈ {0,1}, i.e. if and only ifM is spin orw2(M)≡emod2.
Proof. We claim that the following relation holds:
w2(X) =π∗w2(M).
This follows because the tangent bundle ofX is given by T X = π∗T M ⊕Rand the Whitney sum formula implies w2(T X) = w2(π∗T M)∪w0(R) = π∗w2(T M). Hence X is spin if and only if w2(M)is in the kernel ofπ∗.
We consider the following part of theZ2-Gysin sequence:
H0(M;Z2)−→∪e H2(M;Z2)−→π∗ H2(X;Z2),
wheree denotes the mod 2 reduction ofe. We see that the kernel of π∗ is{0, e}. This implies the claim.
We now specialize to the case where the dimension ofMis equal to 4.
Theorem 9.11. LetM be a simply-connected closed oriented 4-manifold andXthe circle bundle over M with indivisible Euler classe. ThenX is a simply-connected closed oriented 5-manifold and the homology and cohomology ofXare torsion free. We have:
• H0(X;Z)∼=H5(X;Z)∼=Z
• H1(X;Z)∼=H4(X;Z)∼= 0
• H2(X;Z)∼=H3(X;Z)∼=Zb2(M)−1.
Proof. We only have to prove that the cohomology ofXis torsion free and the formula forH2(X;Z).
The cohomology groupsH0(X), H1(X)andH5(X)are always torsion free for an oriented 5-manifold X. We have the following part of the Gysin sequence:
. . .−→H3(M)−→π∗ H3(X)−→π∗ H2(M)−→. . .
By assumption,H3(M) = 0. Therefore the homomorphismπ∗injectsH3(X)intoH2(M), which is torsion free by the assumption thatM is simply-connected. HenceH3(X;Z) is torsion free itself. It remains to considerH2(X)andH4(X). By the Universal Coefficient Theorem and Poincar´e duality, H2(X)is torsion free if and only ifH1(X)is torsion free, if and only ifH4(X)is torsion free. Since H1(X) = 0, we see thatH2(X)andH4(X)are torsion free.
By Lemma 9.9 we haveH2(X;Z) ∼= H2(M;Z)/Z·e. SinceH2(M;Z)is torsion free andeis indivisible,H2(M;Z)/Z·e∼=Zb2(M)−1. This implies the formula forH2(X;Z)∼=H3(X;Z).
By the classification theorem for simply-connected 5-manifolds, in particular Corollary 7.30, we get the following theorem (this has also been proved in [32]).
Theorem 9.12. LetM be a simply-connected closed oriented 4-manifold andXthe circle bundle over M with indivisible Euler classe. ThenXis diffeomorphic to
• X= #(b2(M)−1)S2×S3ifXis spin, and
• X = #(b2(M)−2)S2×S3#S2טS3 ifXis not spin.
The first case occurs if and only ifw2(M)≡αemod2, for someα∈ {0,1}.
Since every closed oriented 4-manifold isSpinc and hence w2(M) is the mod 2 reduction of an integral class, we conclude as a corollary that every closed simply-connected 4-manifoldM is diffeo-morphic to the quotient of a free and smoothS1-action on#(b2(M)−1)S2×S3.