Equivariant path fields on topological manifolds

TL;DR

This paper extends classical results about nowhere vanishing vector fields to an equivariant setting on topological manifolds with group actions, broadening the understanding of manifold structures under symmetry constraints.

Contribution

It provides an equivariant version of Brown's theorem, linking the existence of equivariant path fields to topological and group action properties on manifolds.

Findings

Established an equivariant analog of Brown's theorem for G-manifolds.

Connected the existence of equivariant path fields with topological invariants.

Extended classical manifold results to symmetric settings with finite group actions.

Abstract

A classical theorem of H. Hopf asserts that a closed connected smooth manifold admits a nowhere vanishing vector field if and only if its Euler characteristic is zero. R. Brown generalized Hopf's result to topological manifolds, replacing vector fields with path fields. In this note, we give an equivariant analog of Brown's theorem for locally smooth $G$-manifolds where $G$ is a finite group.