Topological rigidity and H_1-negative involutions on tori

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.

50 pages, to appear in Geometry & Topology

References in corpus (3)

• On the domain of the assembly map in algebraic K-theory
• Algebraic K-theory over the infinite dihedral group: a controlled topology approach
• The surgery obstruction groups of the infinite dihedral group

Cited by in corpus (5)

• Topological rigidity and actions on contractible manifolds with discrete singular set
• On the stable Cannon Conjecture
• A Geometric Interpretation of Ranicki Duality
• Manifolds homotopy equivalent to certain torus bundles over lens spaces
• Countable approximation of topological $G$-manifolds: compact Lie groups $G$