Oferta y cobertura de algunas universidades como parte del dispositivo de

LetXbe a smooth manifold. We consider the long exact sequence

. . .−→H2(X;Z)−→p∗ H2(X;Z₂)−→β H3(X;Z)−→. . .

associated to the short exact sequence of coefficients0 _→ Z →·2 Z →p Z₂ → 0. The homomorphism

β is the associated Bockstein homomorphism andp_∗α ∈H2(X;Z₂)forα ∈H2(X;Z)is called the

mod2reduction ofα. LetE → X be anR-vector bundle. The image of the second Stiefel-Whitney

classw2(E)underβis denoted byW3(E). In particular,W3(E) = 0if and only ifw2(E)is the mod 2 reduction of an integral class.

The existence question for almost contact structures on 5-manifolds was settled by the following theorem of Gray [57].

Theorem 8.16. LetXbe a closed, orientable 5-manifold. ThenXadmits an almost contact structure if and only ifW3(X) = 0.

The existence of contact structures on simply-connected 5-manifolds was proved by Geiges [51]. He also proved a classification theorem for almost contact structures on simply-connected 5-manifolds up to homotopy:

Theorem 8.17. LetXbe a simply-connected, closed 5-manifold.

• Every class inH2(X;Z)that reduces mod2tow₂(X)arises as the first Chern class of an almost

contact structure. Two almost contact structures ξ0, ξ1 are homotopic if and only if c1(ξ0) =

c1(ξ1).

• Every homotopy class of almost contact structures admits a contact structure.

A different proof for the existence of contact structures on simply-connected 5-manifolds can be found in [74, 75]. We will prove the following generalization for the classification of almost contact structures:

VIII.4 Homotopy classification of almost contact structures in dimension 5 161 Theorem 8.18. Let X be a closed, oriented 5-manifold without 2-torsion in H2(X;Z). Then two

almost contact structuresξ0andξ1onXare homotopic if and only ifc1(ξ0) =c1(ξ1).

One direction is clear: if two almost contact structures are homotopic, then they have the same first Chern classes. We now prove the other direction, which requires some preparations.

LetXbe a closed, oriented 5-manifold and(ξ, J)an almost contact structure onX, whereJ is a compatible complex structure onξ. Thenξis the associated vector bundle of a principalU(2)bundle overXthat we denote, for simplicity, also byξ.

There is a principal bundle

F r(X) ←−−−− U(2)

  y

Z

which we callE. HereZdenotes the manifold Fr(X)/U(2)as in Section VIII.2. As seen above,ξcan be thought of as a sectionf of the bundle

Z ←−−−− CP3 =SO(5)/U(2)   y X In fact,ξ =f∗Eas aU(2)-bundle.

We need to determine the first six homotopy groups ofCP3. For this we consider the Hopf fibration

S7 ←−−−− S1

  y

CP3

and the following part of the long exact homotopy sequence for this fibration:

0→0→π5(CP3)→0→0→π₄(CP3)→0→0→π₃(CP3)→0→

0→π2(CP3)→Z→0→π₁(CP3)→0→0.

From this we see that

π2(CP3) =Z

πi(CP3) = 0 i= 0,1,3,4,5

We now consider the following principal bundle

SO(5) ←−−−− U(2)

  y

CP3

which we denote byE. Supposeh:S2 →CP3is a continuous map. Let[h]denote the integer given

by[h]∈π2(CP3)∼=H₂(CP3;Z)∼=Z. We want to prove the following relation:

2[h] =hc1(E), h∗[S2]i

The following part of the long exact homotopy sequence for the bundleE

π2(SO(5))→π2(CP3)→∂ π₁(U(2))→π₁(SO(5))→π₁(CP3),

is given by

0→Z→∂ Z→Z₂ →0.

This shows that∂:π2(CP3) → π₁(U(2))is multiplication by 2 inZ. On the other hand it is known

that

∂h=hc1(E),[h]i for all[h]∈π2(CP3), cf. Lemma 9.7. This implies the claim.

Letξ0, ξ1be two almost contact structures onXgiven by sectionsf0, f1 of theCP3bundleZ →

X. We want to determine whenf0 andf1 are homotopic as sections. Sinceπi(CP3)vanishes in all

degrees less or equal than 5, except for π2(CP4) = Z, the only obstruction comes from degree 2.

Hence we can assume that there exists a homotopyK betweenf0andf1 on the 1-skeletonX(1)and have to see when we can find a homotopy betweenf0andf1 onX(2). This happens if and only if the obstruction classd(f¯ 0, K, f1) ∈H2(X;π2(CP3)) =H2(X;Z)vanishes. The following lemma will

therefore complete the proof of Theorem 8.18.

Lemma 8.19. Ifc1(ξ0) =c1(ξ1), then2 ¯d(f0, K, f1) = 0.

Proof. Letσbe a 2-cell fromX(2). As explained above, we get a map

Fσ =f0|σ×{0}∪K|∂σ×I∪f1|σ×{1}:∂(σ×I)→CP3.

This map determines an element inπ2(CP3)which we denoted byd(f₀, K, f₁)(σ). Sinceπ₁(CP3) =

0, we can homotopFσ such that the domains of f0, f1 are shrunk to smaller 2-cells and K becomes constant. Hence we may assume that f0 andf1 are already identical and constant on X(1) and the homotopyKis a constant map.

The mapsfion the 2-cellσthen induce mapshσ_iˆon the 2-sphereσˆ=σ/∂σ, fori= 0,1. We have

d(f0, K, f1)(σ) = [hσ1ˆ]−[hσ0ˆ].

These maps for all 2-cells inX(2)fit together to give a commutative diagram

X(2) p fi // Z X(2)/X(1) hi : : v v v v v v v v v v Now recall that we have the principal bundleE

F r(X) ←−−−− U(2)   y Z We know that c1(ξi) =c1(fi∗E) =p∗c1(h∗iE)

VIII.5 The level structure of almost contact structures in dimension 5 163