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We compute the additive structure of the RO(C_n)-graded Bredon equivariant cohomology of the equivariant classifying space B_{C_n}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=$. In particular, as an algebra over the cohomology of a point, we show that the cohomology of B_{C_2}SU(2) is generated by two elements subject to a single relation: writing sigma for the sign representation of C_2 in 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}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}SU(2) a satisfactory C_n-cell complex structure.


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