Existence Of Contact Structure

TL;DR

This paper proves that any close almost contact manifold can be homotoped to a genuine contact structure, resolving a question posed by Chern and extending McDuff's theorem.

Contribution

It demonstrates that almost contact structures on closed manifolds are homotopic to contact structures, confirming a longstanding conjecture.

Findings

Homotopy equivalence between almost contact and contact structures

Extension of McDuff's theorem to closed manifolds

Resolution of Chern's question about contact structures

Abstract

Let $(\Sigma ,\xi ',\omega)$ be a close almost contact $(2n-1)$-manifold. Then, by McDuff's theorem, we prove that $\xi '$ is homotopic to a contact structure $\xi $. This answers a question proposed by Chern.