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

Abstract

We discuss a 'Lefschetz filtration' of $\hat{I}{›}^{\ast }({\mathbb{Z}}^{2g})$ 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 $\hat{I}{›}^{\ast }({\mathbb{Z}}^{2g})$, 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 $H{F}^{\infty }(\hat{I}{\pounds }_g\times S^1;\mathbb{Z})$ as modules over the mapping class group. Our computation implies that $H{F}^{\infty }$ is not naturally isomorphic to Mark's 'cup homology'.