Exit path categories induced by group actions

TL;DR

This paper investigates the structure of exit path categories in manifolds with finite group actions, revealing how these categories relate through fibrations and classifying functors, and providing technical tools for lifting homotopies.

Contribution

It establishes that the functor from the exit path category of a manifold to that of its quotient is a right fibration and classifies the enter category via the orbit category, advancing understanding of group actions on manifolds.

Findings

The functor : Exit(M) ; Exit(M/G) is a right fibration.

Enter(M/G) is classified by a functor to the orbit category O_G.

Lifts of homotopies in M/G to M are facilitated by manipulating exit paths.

Abstract

We prove a structural result concerning the exit path category associated to a manifold $M$ equipped with a smooth action of a finite group $G$. Specifically, the functor $\Pi: \mathsf{Exit}(M) \rightarrow \mathsf{Exit}(M/G)$ is a right fibration and $\mathsf{Enter}(M/G)$ is classified by a natural functor $\mathsf{Enter}(M/G) \rightarrow O_G$, where $O_G$ is the orbit category of $G$. The main technical result manipulates exit paths to immediately exiting paths, enabling lifts of homotopies in $M/G$ to homotopies in $M$.