Mod 2 cohomology ring of a kind of orbit configuration space

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.