Contractibility of manifolds by means of stochastic flows

TL;DR

This paper demonstrates that under certain stochastic flow conditions, a manifold's topology is trivial, proving it is contractible by showing all its homotopy groups vanish.

Contribution

It extends previous results by proving that specific stochastic flow conditions imply the manifold is contractible, not just simply connected.

Findings

Manifolds under certain stochastic flows are contractible.

All homotopy groups of the manifold vanish.

The results generalize prior conditions for fundamental group triviality.

Abstract

In the paper [Probab. Theory Relat. Fields, 100 (1994) 417-428] Xue-Mei Li has shown that the moment stability of an SDE is closely connected with the topology of the underlying manifold. In particular, she gave sufficient condition on SDE on a manifold $M$ under which the fundamental group $\pi_1 M=0$. We prove that in fact under the similar conditions the manifold $M$ is contractible, that is all homotopy groups $\pi_k M$, $k\geq1$, vanish.