A Lefschetz decomposition over \mathbb Z, and applications

2025/07/01 by Valiente, Analisa Faulkner, Eismeier, Mike Miller
#14F40 (Secondary) #57K31 (Primary) 20J06 #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2507.00844

Abstract

We discuss a 'Lefschetz filtration' of Λ^*(\mathbb Z^2g) and prove its subquotients are isomorphic as 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 Sp(2g)-module structure on the cohomology of integer Heisenberg groups, and a computation of the Heegaard Floer homology groups HF^∞(Σ_g × S¹; \mathbb Z) as modules over the mapping class group. Our computation implies that HF^∞ is not naturally isomorphic to Mark's 'cup homology'.

Citations

• On integral rigidity in Seiberg-Witten theory
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• Exterior Differential Systems and Euler-Lagrange Partial Differential Equations
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• Holomorphic disks and topological invariants for closed three-manifolds

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