Spherical complexes attached to symplectic lattices

TL;DR

This paper studies the topology of certain posets related to symplectic lattices and their implications for the homology of moduli spaces and symplectic groups, using Cohen-Macaulay properties and nerve theorems.

Contribution

It introduces two Cohen-Macaulay posets associated with Sp(2g,Z) and applies these to topology of abelian varieties and stability of symplectic group homology.

Findings

Posets have Cohen-Macaulay property

Homotopy type of a bouquet of spheres for certain loci

Improved stability range for symplectic group homology

Abstract

To the integral symplectic group Sp(2g,Z) we associate two posets of which we prove that they have the Cohen-Macaulay property. As an application we show that the locus of marked decomposable principally polarized abelian varieties in the Siegel space of genus g has the homotopy type of a bouquet of (g-2)-spheres. This, in turn, implies that the rational homology of moduli space of (unmarked) principal polarized abelian varieties of genus g modulo the decomposable ones vanishes in degree g-2 or lower. Another application is an improved stability range for the homology of the symplectic groups over Euclidean rings. But the original motivation comes from envisaged applications to the homology of groups of Torelli type. The proof of our main result rests on a refined nerve theorem for posets that may have an interest in its own right.