Equivariant path fields on topological manifolds

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.

💡 Research Summary

The paper “Equivariant path fields on topological manifolds” extends the classical Hopf–Brown theorem to the setting of finite‑group actions on locally smooth topological manifolds. Recall that Hopf proved a closed, connected smooth manifold M admits a nowhere‑zero vector field if and only if its Euler characteristic χ(M) vanishes. Brown later showed that the same statement holds for topological manifolds when vector fields are replaced by “path fields”: a continuous assignment x↦γₓ: