Topological rigidity and $H_1$–negative involutions on tori

Abstract

We show, for $n\equiv 0,1\left(mod4\right)$ 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 $n\equiv 2,3\left(mod4\right)$ 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 ${\mathcal{S}}_{TOP}\left({ℝ}^n,{\Gamma }_n\right)$ for the corresponding crystallographic group ${\Gamma }_n$ in terms of the Cappell $UNil$–groups arising from its infinite dihedral subgroups.