  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 $S^3$.
