Is there a higher dimensional analogue to this proposition?

Question

In Geiges' book An Introduction to Contact Topology, there is the following proposition:

Proposition 4.3.2: For any even element $e \in H^2(M,\mathbb{Z})$ there is a contact structure $\xi$ on $M$ with $e(\xi) = e$.

Here $M$ is a 3-dimensional manifold and $e(\xi)$ is the Euler class of the contact structure (I assume we are thinking about it as a bundle).

To what extent is the analogous statement true for higher dimensional manifolds? I.e. if $M$ is an $n+1$ dimensional manifold, is it true that for any even element in $H^n(M,\mathbb{Z})$ there is a contact structure which has that element as its Euler class?

Answer 1

First of all, contact manifolds are odd-dimensional, so the correct generalization of the statement is:

Question. Let $M$ be a closed $(2n+1)$-manifold and $e\in H^{2n}(M)$, is there a contact structure $M$ with Euler class $e$?

In dimension three, the main tools to answer this question are the following:

• the existence of a contact structure on $M$ (Martinet, 1971),
• performing Lutz's twists on a contact structure and understanding its effect on the Euler class.

Therefore, in order to answer this question in all generality, it is first needed to ask whether $M$ carries contact structure and there are homotopical obstructions to this existence problem. In particular, it should be possible to reduce the structure group of $M$ to $U(n)\times\{1\}$, indeed on a contact manifold this reduction is locally given by an almost complex structure on the contact structure and a Reeb direction.

Example. The $5$-manifold $SU(3)/SO(3)$ carries no contact structure (Stong, 1974).

However, this is the only homotopical obstruction for odd-dimensional closed manifolds:

Theorem. (Borman, Eliashberg, Murphy, 2015) Any closed $(2n+1)$-manifold whose structure group can be reduced to $U(n)\times\{1\}$ carries a contact structure.

A better question is then:

Question. Let $M$ be closed $(2n+1)$-manifold whose structure group can be reduced $U(n)\times\{1\}$ and $e\in H^{2n}(M)$, is there a contact structure on $M$ with Euler class $e$?

I have not really think about it, but I strongly believe this is true! A trick similar to Lutz's twists are certainly doable in higher dimension!