The RO(G) Cohomology of Equivariant Classifying Spaces

We compute the $RO(G)$-graded cohomology of various $G$-equivariant classifying spaces, where $G$ is a cyclic 2-group. We then relate these descriptions to ``genuine" $G$-equivariant characteristic class and power operations. Depending on the context, we take coefficients $R$ in the constant Green functors $Z, F_2$ or the rational Burnside Green functor $A_Q$. The classifying spaces we study are $B_GL$ for $L$ a compact connected Lie group such as $U(m),SO(m),Sp(m)$. For certain combinations of $G,L$ and $R$, we compute $\underline{H}_G^{\bigstar}(B_GL;\underline{R})$ as an $RO(G)$-graded Green functor algebra over the cohomology of a point $\underline{H}_G^{\bigstar}(*;\underline{R})$. We also develop a computer program to partially verify and automate these computations. The results in this dissertation first appeared in the author's preprints [3,4,5,6,7].