A Lefschetz decomposition over $\mathbb Z$, and applications

TL;DR

This paper develops an integral Lefschetz decomposition for $igwedge^*(Z^{2g})$, revealing its structure as an $ ext{Sp}(2g)$-module and applying it to cohomology and Floer homology computations.

Contribution

It introduces an integral Lefschetz filtration and demonstrates its applications to cohomology of integer Heisenberg groups and Heegaard Floer homology.

Findings

Failure of Hard Lefschetz over $Z$ clarified

Explicit $ ext{Sp}(2g)$-module structures described

Heegaard Floer homology computed as a mapping class group module

Abstract

We discuss a 'Lefschetz filtration' of $\Lambda^*(\mathbb Z^{2g})$ 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 $\Lambda^*(\mathbb Z^{2g})$, 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 $HF^\infty(\Sigma_g \times S^1; \mathbb Z)$ as modules over the mapping class group. Our computation implies that $HF^\infty$ is not naturally isomorphic to Mark's 'cup homology'.