The existence of contact structures on 9-manifolds

TL;DR

This paper characterizes when closed 9-manifolds admit almost contact and over-twisted contact structures using Stiefel-Whitney classes and homotopy invariants, establishing conditions for existence and invariance under homotopy equivalence.

Contribution

It provides necessary and sufficient conditions for the existence of almost contact and over-twisted contact structures on 9-manifolds based on topological invariants, extending contact topology theory.

Findings

Conditions for almost contact structures in terms of Stiefel-Whitney classes.

Equivalence of over-twisted contact structure existence under homotopy.

Implication that W_3(M)=0 leads to W_7(M)=0.

Abstract

We give necessary and sufficient conditions for a closed orientable 9-manifold M to admit an almost contact structure. The conditions are stated in terms of the Stiefel-Whitney classes of M and other more subtle homotopy invariants of M. By a fundamental result of Borman, Eliashberg and Murphy, M admits an almost contact structure if and only if M admits an over-twisted contact structure. Hence we give necessary and sufficient conditions for M to admit an over-twisted contact structure and we prove that if N is another closed 9-manifold which is homotopy equivalent to M, then M admits an over-twisted contact structure if and only if N does. In addition, for W_i(M) the i-th integral Stiefel-Whitney class of M, we prove that if W_3(M) = 0 then W_7(M) = 0.