Cohomology of Toroidal Orbifold Quotients

TL;DR

This paper explicitly computes the integral cohomology groups of toroidal orbifold quotients induced by cyclic group actions, providing detailed results for crystallographic groups with prime holonomy.

Contribution

It offers a new explicit method for calculating cohomology of toroidal orbifolds under cyclic group actions and applies this to crystallographic groups with prime holonomy.

Findings

Explicit cohomology calculations for toroidal orbifolds

Cohomology of classifying spaces for finite subgroup families

Results applicable to crystallographic groups with prime holonomy

Abstract

Let $\phi:\Z/p\to GL_{n}(\Z)$ denote an integral representation of the cyclic group of prime order $p$. This induces a $\Z/p$-action on the torus $X=\R^{n}/\Z^{n}$. The goal of this paper is to explicitly compute the cohomology groups $H^{*}(X/\Z/p;\Z)$ 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 $\Gamma =\Z^n\rtimes\Z/p$ with prime holonomy.