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Transactions of the American Mathematical Society

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Cohomology of the symplectic group $ {\rm Sp}\sb 4({\bf Z})$. I. The odd torsion case

Authors: Alan Brownstein and Ronnie Lee
Journal: Trans. Amer. Math. Soc. 334 (1992), 575-596
MSC: Primary 11F75; Secondary 11F46, 32G15
MathSciNet review: 1055567
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Abstract: Let $ {h_2}$ be the degree two Siegel space and $ Sp(4,\mathbb{Z})$ the symplectic group. The quotient $ Sp(4,\mathbb{Z})\backslash {h_2}$ can be interpreted as the moduli space of stable Riemann surfaces of genus $ 2$. This moduli space can be decomposed into two pieces corresponding to the moduli of degenerate and nondegenerate surfaces of genus $ 2$. The decomposition leads to a Mayer-Vietoris sequence in cohomology relating the cohomology of $ Sp(4,\mathbb{Z})$ to the cohomology of the genus two mapping class group $ \Gamma _2^0$. Using this tool, the $ 3$- and $ 5$-primary pieces of the integral cohomology of $ Sp(4,\mathbb{Z})$ are computed.

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