Bigraded Poincaré polynomials and the equivariant cohomology of Rep($C_2$)-complexes
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| Format: | Preprint |
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2024
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| _version_ | 1866913733353144320 |
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| author | Hogle, Eric |
| author_facet | Hogle, Eric |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_01117 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bigraded Poincaré polynomials and the equivariant cohomology of Rep($C_2$)-complexes Hogle, Eric Algebraic Topology 55P91, 14F43, 55N25, 55N91, 14M15 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. |
| title | Bigraded Poincaré polynomials and the equivariant cohomology of Rep($C_2$)-complexes |
| topic | Algebraic Topology 55P91, 14F43, 55N25, 55N91, 14M15 |
| url | https://arxiv.org/abs/2410.01117 |