Stratifications of inertia spaces of compact Lie group actions

TL;DR

This paper investigates the topology of inertia spaces arising from compact Lie group actions on manifolds, providing explicit stratifications and a de Rham theorem for differential forms on these spaces.

Contribution

It introduces an explicit Whitney stratification of inertia spaces and proves a de Rham theorem, advancing understanding of their topological and differential structure.

Findings

Inertia space admits a Whitney stratification.

Inertia space is triangulable as a differentiable stratified space.

De Rham theorem established for differential forms on inertia spaces.

Abstract

We study the topology of the inertia space of a smooth $G$-manifold $M$ where $G$ is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differential forms defined on the inertia space with respect to this stratification.