Equivariant smoothing of piecewise linear manifolds

TL;DR

This paper proves that all low-dimensional piecewise linear manifolds with finite group actions can be smoothly structured while preserving the group action, solving longstanding conjectures in topology.

Contribution

It establishes that finite group actions on piecewise linear manifolds up to dimension four can be made smooth, confirming conjectures and solving a challenge posed by Thurston.

Findings

Every piecewise linear manifold of dimension ≤4 admits a compatible smooth structure.

Finite group actions can be made smooth on these manifolds.

Confirms conjectures by Kwasik and Lee, and a challenge by Thurston.

Abstract

We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge posed by Thurston in dimension three and confirms a conjecture by Kwasik and Lee in dimension four in a stronger form.