Cohomology of the symplectic group 𝑆𝑝₄(𝑍). I. The odd torsion case

Brownstein, Alan; Lee, Ronnie

doi:10.1090/S0002-9947-1992-1055567-X

Cohomology of the symplectic group $\textrm {Sp}_ 4(\textbf {Z})$. I. The odd torsion case
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by Alan Brownstein and Ronnie Lee

Trans. Amer. Math. Soc. 334 (1992), 575-596

DOI: https://doi.org/10.1090/S0002-9947-1992-1055567-X

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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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