Topological rigidity and H_1-negative involutions on tori

TL;DR

This paper classifies involutions on n-tori acting as -Id on first homology, revealing unique cases and infinitely many with specific fixed point properties, using equivariant topology and algebraic K-theory techniques.

Contribution

It provides a complete classification of certain involutions on tori, including their conjugacy classes and fixed point counts, through explicit computation of equivariant structure sets.

Findings

Unique involution for specific n values

Infinitely many involutions with 2^n fixed points

Complete equivariant topological rigidity analysis

Abstract

We prove there is only one involution (up to conjugacy) on the n-torus which acts as $-\mathrm{Id}$ on the first homology group when $n$ is of the form $4k$, is of the form $4k+1$, or is less than $4$. In all other cases we prove there are infinitely many such involutions up to conjugacy, but each of them has exactly $2^n$ fixed points and is conjugate to a smooth involution. The key technical point is that we completely compute the equivariant structure set for the corresponding crystallographic group action on $\mathbb{R}^n$ in terms of the Cappell $\mathrm{UNil}$-groups arising from its infinite dihedral subgroups. We give a complete analysis of equivariant topological rigidity for this family of groups.