Isovariant homotopy theory and fixed point invariants

TL;DR

This paper develops an isovariant homotopy theory framework for finite group actions, linking fixed point removal to Reidemeister trace vanishing, and extends Whitehead's theorem to the isovariant setting.

Contribution

It introduces an isovariant analogue of equivariant intersection theory, establishing conditions for fixed point removal and formulating an isovariant Whitehead's theorem.

Findings

Fixed points can be removed isovariantly when Reidemeister trace vanishes.

Established an isovariant Whitehead's theorem for manifolds.

Provided insights into distinguishing manifolds via isovariant homotopy theory.

Abstract

An isovariant map is an equivariant map between $G$-spaces which strictly preserves isotropy groups. We consider an isovariant analogue of Klein--Williams equivariant intersection theory for a finite group $G$. We prove that under certain reasonable dimension and codimension conditions on $H$-fixed subspaces (for $H\leq G$), the fixed points of a self-map of a compact smooth $G$-manifold can be removed isovariantly if and only if the equivariant Reidemeister trace of the map vanishes. In doing so, we study isovariant maps between manifolds up to isovariant homotopy, yielding an isovariant Whitehead's theorem. In addition, we speculate on the role of isovariant homotopy theory in distinguishing manifolds up to homeomorphism.