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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.00844 |
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| _version_ | 1866916819822968832 |
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| author | Valiente, Analisa Faulkner Eismeier, Mike Miller |
| author_facet | Valiente, Analisa Faulkner Eismeier, Mike Miller |
| contents | We discuss a 'Lefschetz filtration' of $Λ^*(\mathbb Z^{2g})$ and prove its subquotients are isomorphic as $\text{Sp}(2g)$-modules to primitive subspaces $P^k(\mathbb Z^{2g})$. This gives a sort of integral version of the Lefschetz decomposition over $\mathbb C$.
We present three applications: the precise failure of the Hard Lefschetz theorem for $Λ^*(\mathbb Z^{2g})$, a description of the $\text{Sp}(2g)$-module structure on the cohomology of integer Heisenberg groups, and a computation of the Heegaard Floer homology groups $HF^\infty(Σ_g \times S^1; \mathbb Z)$ as modules over the mapping class group. Our computation implies that $HF^\infty$ is not naturally isomorphic to Mark's 'cup homology'. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00844 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Lefschetz decomposition over $\mathbb Z$, and applications Valiente, Analisa Faulkner Eismeier, Mike Miller Geometric Topology Algebraic Geometry 57K31 (Primary) 20J06, 14F40 (Secondary) We discuss a 'Lefschetz filtration' of $Λ^*(\mathbb Z^{2g})$ and prove its subquotients are isomorphic as $\text{Sp}(2g)$-modules to primitive subspaces $P^k(\mathbb Z^{2g})$. This gives a sort of integral version of the Lefschetz decomposition over $\mathbb C$. We present three applications: the precise failure of the Hard Lefschetz theorem for $Λ^*(\mathbb Z^{2g})$, a description of the $\text{Sp}(2g)$-module structure on the cohomology of integer Heisenberg groups, and a computation of the Heegaard Floer homology groups $HF^\infty(Σ_g \times S^1; \mathbb Z)$ as modules over the mapping class group. Our computation implies that $HF^\infty$ is not naturally isomorphic to Mark's 'cup homology'. |
| title | A Lefschetz decomposition over $\mathbb Z$, and applications |
| topic | Geometric Topology Algebraic Geometry 57K31 (Primary) 20J06, 14F40 (Secondary) |
| url | https://arxiv.org/abs/2507.00844 |