# Mod 2 cohomology ring of a kind of orbit configuration space

## TL;DR

This paper computes the mod 2 cohomology ring of a specific orbit configuration space related to small covers, using a differential graded algebra approach, and extends to real arrangement complements.

## Contribution

It introduces a differential graded algebra method to determine the mod 2 cohomology ring of orbit configuration spaces, providing a new computational tool.

## Key findings

- Established a ring isomorphism between the algebra and the cohomology ring.
- Provided a method applicable to real arrangement complement spaces.
- Enhanced understanding of the topology of orbit configuration spaces.

## Abstract

In this paper we caculate mod 2 cohomology ring of $F_{\mathbb{Z}_2^m}(\mathbb{R}^m,n)$ , which is local representation of orbit congfiguration spaces over small covers. We construct a differntial graded algebra, and there is a ring isomorphism between its mod 2 cohomology ring and $H^*(F_{\mathbb{Z}_2^m}(\mathbb{R}^m,n),\mathbb{Z}_2)$. This idea can also be applied to calculate mod 2 cohomology ring of complement space of real arrangements.