The $\mathrm{RO}(G)$-Graded Cohomology of the Equivariant Classifying Space $B_G\mathrm{SU}(2)$

Abstract

We compute the additive structure of the $\mathrm{RO}(C_n)$-graded Bredon equivariant cohomology of the equivariant classifying space $B_{C_n}\mathrm{SU}(2)$, for any $n$ that is either prime or a product of distinct odd primes, and we also compute its multiplicative structure for $n=2$. In particular, as an algebra over the cohomology of a point, we show that the cohomology of $B_{C_2}\mathrm{SU}(2)$ is generated by two elements subject to a single relation: writing $\sigma$ for the sign representation of $C_2$ in $\mathrm{RO}(C_2)$, the generators are an element $c$ in dimension $4\sigma$ and an element $C$ in dimension $4+4\sigma$, satisfying the relation $c^2 = \epsilon^4 c + \xi^2 C$, where $\epsilon$ and $\xi$ are elements of the cohomology of a point. Throughout, we take coefficients in the Burnside ring Mackey functor $A$. The key tools used are equivariant "even-dimensional freeness" and "multiplicative comparison" theorems for $G$-cell complexes, both proven by Lewis in [Lew88] and subsequently refined by Shulman in [Shu10], and with the former theorem extended by Basu and Ghosh in [BG16]. The latter theorem enables us to compute the multiplicative structure of the cohomology of $B_{C_2}\mathrm{SU}(2)$ by embedding it in a direct sum of cohomology rings whose structure is more easily understood. Both theorems require the cells of the $G$-cell complex to be attached in a well-behaved order, and a significant step in our work is to give $B_{C_n}\mathrm{SU}(2)$ a satisfactory $C_n$-cell complex structure.