Geometriae Dedicata

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

Spherical complexes attached to symplectic lattices

Open Access
Original Paper


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.


Integral symplectic group Cohen-Macaulay poset 

Mathematics Subject Classification (2010)

05E18 11E57 19B14 


  1. 1.
    Brown K.S.: Buildings, p. vii + 215. Springer, New York (1989)MATHGoogle Scholar
  2. 2.
    Charney R.: On the problem of Homology Stability for Congruence subgroups. Comm. Algebra 12(17), 2081–2123 (1984)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Charney R.: A generalization of a theorem of Vogtmann. J. Pure Appl. Algebra 44, 107–125 (1987)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Debarre, O.: Complex Tori and Abelian Varieties. SMF/AMS Text Monogr. 11, x+109, Paris (2005)Google Scholar
  5. 5.
    Looijenga, E.: Connectivity of Complexes of Separating Curves. \({{\tt arXiv:1001.0823}}\) Google Scholar
  6. 6.
    Maazen, H.: Homology Stability for the General Linear Group. Thesis Utrecht (1979). Available at
  7. 7.
    Mirzaii B., van der Kallen W.: Homology stability for unitary groups. Doc. Math. 7, 143–166 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Quillen D.: Homotopy properties of the poset of nontrivial p-subgroups of a group. Adv. Math. 28, 101–128 (1978)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    van den Berg, B.: On the Abelianization of the Torelli Group. Thesis Utrecht. Available at (2003)
  10. 10.
    van der Kallen W.: Homology stability for Linear Groups. Invent. Math. 60, 269–295 (1980)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands

Personalised recommendations