Bigraded Poincar\'e polynomials and the equivariant cohomology of Rep($C_2$)-complexes

TL;DR

This paper develops a new computational approach for Bredon cohomology of equivariant spaces, specifically Grassmannians with $C_2$-actions, using a novel statistic on modules to simplify complex calculations.

Contribution

It introduces a new statistic on $ ext{M}_2$-modules that makes cohomology computations for $ ext{Rep}(C_2)$-complexes more feasible, with new results for Grassmannians.

Findings

New statistic simplifies cohomology calculations.

Computed cohomology for specific Grassmannian examples.

Enhanced understanding of $ ext{Rep}(C_2)$-space structures.

Abstract

We are interested in computing the Bredon cohomology with coefficients in the constant Mackey functor $\underline{ \mathbb{F}_2}$ for equivariant $\text{Rep}(C_2)$ spaces, in particular for Grassmannian manifolds of the form $\\text{Gr}_k(V)$ where $V$ is some real representation of $C_2$. It is possible to create multiple distinct $\text{Rep}(C_2)$ constructions of (and hence multiple filtration spectral sequences for) a given Grassmannian. For sufficiently small examples one may exhaustively compute all possible outcomes of each spectral sequence and determine if there exists a unique common answer. However, the complexity of such a computation quickly balloons in time and memory requirements. We introduce a statistic on $\mathbb{M}_2$-modules valued in the polynomial ring $\mathbb{Z}[x,y]$ which makes cohomology computation of Rep($C_2$)-complexes more tractable, and we present some new results for Grassmannians.