Equivariant Cohomology of Projective Spaces

TL;DR

This paper computes the equivariant homology and cohomology of projective spaces under cyclic group actions, revealing decompositions and explicit formulas that advance understanding in equivariant topology.

Contribution

It provides new decompositions of equivariant (co)homology of projective spaces for cyclic groups, including explicit formulas and implications for the slice tower.

Findings

Decomposition of equivariant homology as wedges of suspensions of $H\underline{\mathbb{Z}}$

Explicit formulas for cohomology with $\underline{\mathbb{Z}_p}$ coefficients

Degeneration of the slice tower in these cases

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 $S^3$.