Topological rigidity and H_1-negative involutions on tori

Frank Connolly; James F. Davis; Qayum Khan

doi:10.2140/gt.2014.18.1719

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arXiv · 1102.2660

Topological rigidity and H_1-negative involutions on tori

Frank Connolly·James F. Davis·Qayum Khan

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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.

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BibTeX RIS

Frank Connolly, James F. Davis, Qayum Khan. 2014-02-07. Topological rigidity and H_1-negative involutions on tori. https://doi.org/10.2140/gt.2014.18.1719

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