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

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'.