Geometriae Dedicata

, Volume 152, Issue 1, pp 197–211 | Cite as

Spherical complexes attached to symplectic lattices

Open Access
Original Paper
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Abstract

To the integral symplectic group \({{\rm Sp}(2g,\mathbb{Z})}\) 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. 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.

Keywords

Integral symplectic group Cohen-Macaulay poset 

Mathematics Subject Classification (2010)

05E18 11E57 19B14 

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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands

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