VII.6 Connected sum decomposition of simply-connected 5-manifolds 153
• Q1, . . . , Qnare simply-connected irreducible spin 5-manifolds.
• IfXis spin thenP isS5.
• IfXis not spin thenP is eitherW,X1 = W#W or a simply-connected irreducible non-spin 5-manifoldXkwithk≥2.
Proof. We first prove Uniqueness: Suppose that there exists a diffeomorphism between closed, simply-connected 5-manifolds of the form
X∼=Q1#. . .#Qn#P ∼=Q01#. . .#Q0m#P0.
IfX is spin, then all summands in X have to be spin. This impliesP = P0 = S5. The manifolds Qi, Q0j are of the formMpk for primespand integersk≥1orS2×S3. SinceH2(Mpk;Z)is always torsion andH2(S2×S3;Z)∼=Z, the number ofS2×S3’s among theQi, Qjmust be equal to the rank ofH2(X;Z). Writing the torsion subgroup of H2(X;Z) as a sum of cyclic groups of prime power order determines theMpk summands among theQi, Qj uniquely. This proves the uniqueness claim if Xis spin.
Suppose thatXis not spin. We can find aw2-basis for
H2(X;Z) =H2(Q1;Z)⊕. . .⊕H2(Qn;Z)⊕H2(P;Z)
which is non-zero only on one basis element inH2(P;Z). Hencei(X) = i(P). This determinesP if i(P)≥2. Ifi(P) = 1, thenP is diffeomorphic toW orX2. The sum of the torsion subgroups of the second homology forQ1, . . . , Qnis of the formH⊕H, whereH is a direct sum of groups of prime power order. Hence
TorH2(X;Z)∼=H⊕Hor ∼=H⊕H⊕Z2,
ifP =X2orP =W, respectively. Therefore, TorH2(X;Z)determines whetherP =X2orP =W. This shows that the non-spin summandPis uniquely determined byX, which impliesP ∼=P0.
The number ofS2×S3 is again equal to the rank ofH2(X;Z), ifP 6=S2טS3, and to the rank minus 1, ifP =S2טS3. Since TorH2(P;Z)is already determined, the remaining summandsQi, Qj of the formMpk are determined by TorH2(X;Z). This proves uniqueness of the decomposition ifX is non-spin.
We now prove Existence: Let X be a closed, simply-connected 5-manifold with linking form b. Suppose thati(X) < ∞. All possible linking forms given by Theorem 7.9 can be realized by a connected sum of manifolds of the typeXk, Mpk, W, where only oneXk orW summand is needed.
This follows becausep can by any prime andk ≥ 1any integer. We get a closed, simply-connected 5-manifoldX0with
H2(X0;Z)∼=TorH2(X;Z), i(X0) =i(X).
LetX00=X#rS2×S3, whererdenotes the rank ofH2(X;Z). Then H2(X00;Z)∼=H2(X;Z), i(X00) =i(X).
The closed, simply-connected 5-manifoldX00is of the form as in the statement of the theorem and by Corollary 7.20,XandX00are diffeomorphic.
Suppose that i(X) = ∞. By Corollary 7.10, the torsion subgroup ofH2(X;Z)has to be of the formH⊕H. We can realize this direct sum as the torsion subgroup of a connected sum of manifolds of typeMpk. We add oneS2טS3summand to get a closed, simply-connected 5-manifoldX0with
H2(X0;Z)∼=TorH2(X;Z)⊕Z, i(X0) =i(X).
LetX00X#(r−1)S2×S3, whererdenotes again the rank ofH2(X;Z). Then H2(X00;Z)∼=H2(X;Z), i(X00) =i(X).
HenceXandX00are diffeomorphic by Corollary 7.20.
The following corollary will be used in Chapter IX.
Corollary 7.30. LetXbe a simply-connected closed oriented 5-manifold withH2(X;Z)∼=Zk. Then Xis diffeomorphic to
• #kS2×S3ifXis spin, and
• #(k−1)S2×S3#S2טS3 ifXis not spin.
Proof. This follows from Theorem 7.29 becauseH2(X;Z)is torsion free.
Chapter VIII
Contact structures on 5-manifolds
Contents
VIII.1 Basic definitions . . . . 155 VIII.2 Almost contact structures as sections of a fibre bundle . . . . 158 VIII.3 Overview of obstruction theory . . . . 159 VIII.4 Homotopy classification of almost contact structures in dimension 5 . . . . 160 VIII.5 The level structure of almost contact structures in dimension 5 . . . . 163
In this chapter we recall the basic notions related to contact structures. We then focus on the 5-dimensional case and show that a theorem of H. Geiges [51] on the classification of almost contact structures up to homotopy on simply-connected 5-manifolds can be extended to all 5-manifolds X whoseH2(X;Z)does not contain 2-torsion. In the last section, we show how to classify almost contact structures on simply-connected 5-manifoldsXup to equivalence, where a combination of homotopies and orientation preserving self-diffeomorphisms is allowed. The proof uses Barden’s classification theorem for simply-connected 5-manifolds from Chapter VII, in particular the possibility to realize certain automorphisms ofH2(X;Z)by an orientation preserving self-diffeomorphism ofX.
VIII.1 Basic definitions
LetX2n+1be connected, oriented manifold of odd dimension. Supposeα∈Ω1(X)is a 1-form onX without zeroes. Then
ξ=kerα={(p, v)∈T X |αp(v) = 0}
is a smooth distribution onX(a subbundle ofT X) of rank2n, sinceαpis a non-vanishing linear map TpX→Rfor allp∈X. We consider the 2-formdαonX. It defines a skew-symmetric bilinear form on each tangent spaceTpX.
Definition 8.1. If the restriction(dα)|ξ is symplectic (i.e. non-degenerate), then we callα a contact form. The hyperplane distributionξis called the underlying contact structure.
Every contact form induces an orientation on the contact structureξ, as a vector bundle, through the symplectic form(dα)|ξ. SinceT X is an oriented vector bundle by assumption, the quotientT X/ξ is an oriented real vector bundle of rank 1 and hence trivial. Therefore, we can write
T X =R⊕ξ, (8.1)
whereRdenotes the trivial real vector bundle of rank 1, realized as a subbundle ofT X. Sinceξis the kernel ofα, the 1-formαis non-zero on each non-zero vector ofR.
We have the following equivalent characterization of contact forms.
Lemma 8.2. A 1-formαonXis a contact form if and only ifα∧(dα)nis a volume form onX.
Proof. Supposeαis a contact form. Then(dα)nrestricted toξis a volume form on each fibre. Choose a basise1, . . . , e2n+1ofTpXsuch thatα(e1)6= 0ande2, . . . , e2n+1define an oriented basis ofξ. Then
α∧(dα)n(e1, . . . , e2n+1) =α(e1)(dα)n(e2, . . . , e2n+1)6= 0.
Henceα∧(dα)nis non-zero at each pointp∈Xand therefore a volume form onX.
Conversely, suppose thatα∧(dα)nis a volume form. Lete1, e2, . . . , e2n+1be a basis ofTpXsuch thatα(e1)6= 0ande2, . . . , e2n+1form a basis ofξ =kerα. Since volume forms are always non-zero on bases, the calculation above shows that (dα)n(e2, . . . , e2n+1) 6= 0. This is equivalent to (dα)|ξ
being symplectic.
Since X was assumed oriented to start with, we can compare the orientation of X defined by α∧(dα)nwith the given one.
Definition 8.3. A contact formαis called positive or negative, depending on whether the orientation ofXcoincides with the orientation defined byα∧(dα)n.
Iff: X→ Ris a smooth, nowhere vanishing function andαa contact form onX, thenα0 :=f α andαhave the same kernelξ. Moreover,
dα0 =df∧α+f dα.
Hence(dα0)|ξ=f(dα)|ξis symplectic andα0is also a contact form.
Conversely, suppose that two contact forms α, α0 have the same underlying contact structure ξ.
Then there exists a smooth, nowhere vanishing function f: X → Rsuch that α0 = f α: We may choose a fixed complement Rof ξ inT X, such that α andα0 are both non-zero on each non-zero vector inR. SinceRis trivial, we can choose a nowhere zero sectionv. Then
f(p) := α0p(v) αp(v)
is a well defined smooth, nowhere vanishing function onX. This impliesα0=f α, since this equation holds on the common kernelξand on the sectionv, spanningR. We conclude:
Lemma 8.4. Two 1-formsα, α0 are contact with the same underlying contact structureξ if and only if there exists a smooth, nowhere vanishing functionf:X → Rsuch thatα0 = f α. The symplectic structure induced byα0onξis of the form(dα0)|ξ=f(dα)|ξ.
Letαbe a 1-form. We set
kerdα={(p, v)∈T X |dα(v, x) = 0for allx∈TpX}. Suppose thatαis a contact form. Letv∈TpXbe a vector in kerα∩kerdα. Then
α∧(dα)n(v, v2, . . . , v2n+1) = 0,
for all vectorsv2, . . . , v2n+1inTpX. Sinceα∧(dα)nis a volume form,vhas to be zero.
The 2-formdαcannot be symplectic onX, sinceXis of odd dimension. Hence the kernel ofdα cannot be zero at any pointp ∈X. Since kerdα∩kerα= 0and kerαhas rank2n, the kernel ofdα must be 1-dimensional. IfRis a non-zero element in kerdαthenα(R) 6= 0. Therefore we can make the following definition.
VIII.1 Basic definitions 157 Definition 8.5. Letαbe a contact form onX. Then there exists a unique vector fieldRαonXwith
dα(Rα) = 0, α(Rα) = 1.
Rαis called the Reeb vector field ofα.
The vector fieldRαdefines a splitting
T X =RRα⊕ξ,
as in equation 8.1. However, nowξis not only the kernel ofα, butRRαis the kernel ofdα.
By the Cartan formula,
LRαα = diRαα+iRαdα
= 0.
Hence the flow of the Reeb vector field preserves the contact form and the contact structure.
Letξbe a contact structure. We fix a splittingT X =R⊕ξand a coorientation, i.e. an orientation on the line bundleR. We now only consider defining 1-formsαwhich evaluate positively on the vector defining the orientation onR. There exists a complex structureJ onξcompatible with the symplectic structure (dα)|ξ on each fibre (see Section II.2 in the preliminaries). For fixed (dα)|ξ, the space of such J is contractible, hence we get well-defined Chern classes ck(ξ) ∈ H2k(X;Z), independent of the choice of a compatible J. Moreover, if we choose a different defining form α0 for ξ which evaluates positively on the orientation of R, then by Lemma 8.4 there exists a functionf:X → R which is everywhere positive and such thatα0 =f α. The functionf can be deformed smoothly into the constant function with value1, without ever crossing zero. This implies that the Chern classes do not depend on the choice of defining formα.
Definition 8.6. The Chern classes ck(ξ) ∈ H2k(X;Z) of a cooriented contact structure ξ are well defined and independent of the choice of the defining formα, respecting the coorientation, and the almost complex structureJ.
A contact structure determines, in particular, a symplectic subbundle ofT X of corank1. This is also known as an almost contact structure.
Definition 8.7. An almost contact structure onX2n+1 is a rank 2n-distributionξ with a symplectic structureσonξ.
Clearly, every contact structure determines an almost contact structure. The converse is true if and only if the symplectic structureσonξis of the form(dα)|ξfor a 1-formαonXdefiningξ. For each almost contact structureξ, we can choose again a compatible almost complex structureJ. The space of suchJ is contractible, hence we get well-defined Chern classes. However, they will depend in general on the symplectic structureσ, not only on the distributionξas in the contact case. The first Chern class ofξis related to the second Stiefel-Whitney class in a similar way as in the almost complex case:
Lemma 8.8. Letξbe an almost contact structure onX. Thenc1(ξ)≡w2(M)mod2.
Proof. By the Whitney sum formula forT X =ξ⊕R,
w2(X) =w2(ξ)∪w0(R) =w2(ξ).
Sinceξ →Xis a complex vector bundle, with complex structure compatible withσ, we havew2(ξ)≡ c1(ξ)mod2. This implies the claim.
Suppose thatξt, t∈[0,1]is a smooth family of contact structures on a closed manifoldX. We can choose a smooth family of1-formsαtdefiningξt. Using the Moser technique, one can prove that there exists a smooth familyψt of self-diffeomorphisms ofX withψ0 = IdX such thatψ∗αt = ftα0, for smooth functionsftonX[96]. This implies the following theorem of Gray [57].
Theorem 8.9. Letξt, t∈[0,1]be a smooth family of contact structures on a closed manifoldX. Then there exists an isotopyψt, t∈[0,1]of diffeomorphisms ofXsuch thatψ∗tξt=ξ0.
Because of this theorem, we call contact structures ξ, ξ0 which can be deformed into each other by a smooth family of contact structures isotopic. We call almost contact structures homotopic, if they can be connected by a smooth family of almost contact structures. The contact structures in an isotopy class or the almost contact structures in a homotopy class all have the same Chern classes.
We can also consider (almost) contact structuresξ, ξ0which are permuted by an orientation-preserving self-diffeomorphismψofX, in the sense thatψ∗ξ0 =ξ.
Definition 8.10. We call almost contact structures and contact structures on an oriented manifoldX equivalent, if they can be made identical by a combination of deformations (homotopies, resp. iso-topies) and by orientation-preserving self-diffeomorphisms ofX.
See Vidussi’s article [140] for a related definition for symplectic forms.