On $\mathop{\mathrm{RO}}(G)$–graded equivariant “ordinary” cohomology where $G$ is a power of $\mathbb{Z}/2$

Abstract

We compute the complete $RO\left(G\right)$–graded coefficients of “ordinary” cohomology with coefficients in $\mathbb{Z}∕2$ for $G={\left(\mathbb{Z}∕2\right)}^n$. As an important intermediate step, we identify the ring of coefficients of the corresponding geometric fixed point spectrum, revealing some interesting algebra. This is a first computation of its kind for groups which are not cyclic $p$–groups.