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We compute the additive structure of the $\mathrm{RO}(C_n)$-graded Bredonequivariant cohomology of the equivariant classifying space$B_{C_n}\mathrm{SU}(2)$, for any $n$ that is either prime or a product ofdistinct odd primes, and we also compute its multiplicative structure for$n=2$. In particular, as an algebra over the cohomology of a point, we showthat the cohomology of $B_{C_2}\mathrm{SU}(2)$ is generated by two elementssubject 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 thecohomology of a point. Throughout, we take coefficients in the Burnside ringMackey functor $A$.  The key tools used are equivariant "even-dimensional freeness" and"multiplicative comparison" theorems for $G$-cell complexes, both proven byLewis in [Lew88] and subsequently refined by Shulman in [Shu10], and with theformer theorem extended by Basu and Ghosh in [BG16]. The latter theorem enablesus to compute the multiplicative structure of the cohomology of$B_{C_2}\mathrm{SU}(2)$ by embedding it in a direct sum of cohomology ringswhose structure is more easily understood. Both theorems require the cells ofthe $G$-cell complex to be attached in a well-behaved order, and a significantstep in our work is to give $B_{C_n}\mathrm{SU}(2)$ a satisfactory $C_n$-cellcomplex structure.

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