Equivariant Cohomology of Projective Spaces

Abstract: We compute the equivariant homology and cohomology of projective spaces with integer coefficients. More precisely, in the case of cyclic groups, we show that the cellular filtration of the projective space $P(k\rho )$, of lines inside copies of the regular representation, yields a splitting of $H\underline{\mathbb{Z}}\bigwedge P(k\rho )_+$ as a wedge of suspensions of $H\underline{\mathbb{Z}}$. This is carried out both in the complex case, and also in the quaternionic case, and further, for the $C_2$ action on $\mathbb{C} P^n$ by complex conjugation. We also observe that these decompositions imply a degeneration of the slice tower in these cases. Finally, we describe the cohomology of the projective spaces when $|G|=p^m$ of prime power order, with explicit formulas for $\underline{\mathbb{Z}_p}$-coefficients. Letting $k=\infty$, this also describes the equivariant homology and cohomology of the classifying spaces of $S^1$ and $S^3$.