Topological rigidity and H_1-negative involutions on tori

We prove there is only one involution (up to conjugacy) on the n-torus which acts as $-\mathrm{I}\mathrm{d}$ 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 $\mathrm{U}\mathrm{N}\mathrm{i}\mathrm{l}$-groups arising from its infinite dihedral subgroups. We give a complete analysis of equivariant topological rigidity for this family of groups.