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arXiv · 2410.01117

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

Eric Hogle
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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.

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Eric Hogle. 2024-10-01. Bigraded Poincar\'e polynomials and the equivariant cohomology of Rep($C_2$)-complexes. https://arxiv.org/abs/2410.01117

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