Exit path categories induced by group actions
Patrick Mayeda
TL;DR
The paper extends the classical path-lifting picture from covering spaces to the setting of group actions with fixed points by introducing exit/enter path categories for stratified spaces. It proves that, for a finite group $G$ acting smoothly on a manifold $M$, the projection $\Pi: \mathsf{Exit}(M) \to \mathsf{Exit}(M/G)$ is a right fibration and identifies $\mathsf{Enter}(M/G)$ with a classifying functor to the orbit category $\mathcal{O}_G$, leading to a pullback description of $\mathsf{Enter}(M)$. The key technical developments include lifting exit paths and homotopies across strata, using conically smooth stratifications and fiberwise cones to peel deformations off strata, and replacing general homotopies with immediately exiting ones to control lifts. This yields a canonical pullback square that generalizes the classical covering-space classification via $BG$ to the orbit-category setting, recovering fundamental-groupoid data in the free-action case and offering a structural framework for equivariant exit-path theory. The results provide a robust bridge between stratified topology, equivariant homotopy theory, and higher-categorical notions of fibrations, with potential impact on exodromy and equivariant stratified approaches to space-level classifications.
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 $Π: \mathsf{Exit}(M) \rightarrow \mathsf{Exit}(M/G)$ 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 $M$.