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Cohomology of the Symplectic Group Sp4(Z) Part I: The Odd Torsion Case

Alan Brownstein and Ronnie Lee
Transactions of the American Mathematical Society
Vol. 334, No. 2 (Dec., 1992), pp. 575-596
DOI: 10.2307/2154473
Stable URL: http://www.jstor.org/stable/2154473
Page Count: 22
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Cohomology of the Symplectic Group Sp4(Z) Part I: The Odd Torsion Case
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Abstract

Let h2 be the degree two Siegel space and Sp(4, Z) the symplectic group. The quotient $\operatorname{Sp}(4, \mathbb{Z})\backslash h_2$ 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 $\operatorname{Sp}(4, \mathbb{Z})$ to the cohomology of the genus two mapping class group Γ0 2. Using this tool, the 3- and 5-primary pieces of the integral cohomology of $\operatorname{Sp}(4, \mathbb{Z})$ are computed.

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