#### Volume 18, issue 3 (2014)

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Topological rigidity and $H_1$–negative involutions on tori

### Frank Connolly, James F Davis and Qayum Khan

Geometry & Topology 18 (2014) 1719–1768
##### Abstract

We show, for $n\equiv 0,1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}4\right)$ or $n=2,3$, there is precisely one equivariant homeomorphism class of ${C}_{2}$–manifolds $\left({N}^{n},{C}_{2}\right)$ for which ${N}^{n}$ is homotopy equivalent to the $n$–torus and ${C}_{2}=\left\{1,\sigma \right\}$ acts so that ${\sigma }_{\ast }\left(x\right)=-x$ for all $x\in {H}_{1}\left(N\right)$. If $n\equiv 2,3\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}4\right)$ and $n>3$, we show there are infinitely many such ${C}_{2}$–manifolds. Each is smoothable with exactly ${2}^{n}$ fixed points.

The key technical point is that we compute, for all $n\ge 4$, the equivariant structure set ${\mathsc{S}}_{TOP}\left({ℝ}^{n},{\Gamma }_{n}\right)$ for the corresponding crystallographic group ${\Gamma }_{n}$ in terms of the Cappell $UNil$–groups arising from its infinite dihedral subgroups.

##### Keywords
equivariant rigidity, torus, surgery
Primary: 57S17
Secondary: 57R67