Smooth shifts along flows

TL;DR

This paper investigates the topological structure of certain groups of smooth maps and diffeomorphisms preserving a flow on a manifold, establishing conditions under which these groups are homotopically equivalent or contractible.

Contribution

It proves that, under mild conditions, the identity components of these groups are homotopy equivalent or contractible, clarifying their topological nature.

Findings

Inclusion of diffeomorphisms into smooth maps is a homotopy equivalence under certain conditions.

The spaces are either contractible or homotopy equivalent to a circle.

Results depend on fixed point conditions of the flow.

Abstract

Let $\Phi$ be a flow on a smooth, compact, finite-dimensional manifold $M$. Consider the subsets $E(\Phi)$ and $D(\Phi)$ of $C^{\infty}(M,M)$ consisting of smoothh mappings and diffeomorphisms (respectively) of $M$ preserving the foliation of the flow $\Phi$. Let also $E_{0}(\Phi)$ and $D_{0}(\Phi)$ be the identity path components of $E(\Phi)$ and $D(\Phi)$ with compact-open topology. We prove that under mild conditions on fixed points of $\Phi$ the inclusion $D_{0}(\Phi) \subset E_{0}(\Phi)$ is a homotopy equivalence and these spaces are either contractible or homotopically equivalent to the circle.