Cohomology of Toroidal Orbifold Quotients

Abstract

Let $\varphi :\mathbb{Z}/p\to G{L}_n(\mathbb{Z})$ denote an integral representation of the cyclic group of prime order $p$. This induces a $\mathbb{Z}/p$-action on the torus $X={\mathbb{R}}^n/{\mathbb{Z}}^n$. The goal of this paper is to explicitly compute the cohomology groups $H^{\ast }(X/\mathbb{Z}/p;\mathbb{Z})$ for any such representation. As a consequence we obtain an explicit calculation of the integral cohomology of the classifying space associated to the family of finite subgroups for any crystallographic group $\Gamma ={\mathbb{Z}}^n⋊\mathbb{Z}/p$ with prime holonomy.

arXiv commentFinal version. Accepted for publication in the Journal of Algebra