On \(\mathrm{RO}(G)\)-graded equivariant ``ordinary'' cohomology where \(G\) is a power of \(\mathbb{Z}/2\) (Q518814)

Let \(G=(\mathbb{Z}/2)^n\) and let \(H \mathbb{Z}/2\) be the `constant' \(G\)-Mackey functor for \(\mathbb{Z}/2\). This paper gives a calculation of the \(\mathrm{RO}(G)\)-graded coefficients of \(H \mathbb{Z}/2\). An important intermediary step is to calculate the coefficients of \(\Phi^G H \mathbb{Z}/2\), the \(G\)-geometric fixed points of \(H \mathbb{Z}/2\). This is performed by spectral sequence methods and the resulting ring is identified as a known object from algebraic geometry coming from hyperplane arrangements. The main result is then proven by developing a new spectral sequence to compare the coefficients of \(\Phi^G H \mathbb{Z}/2\) with those of \(H \mathbb{Z}/2\).