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

Abstract

Let h 2 {h_2} be the degree two Siegel space and S p ( 4 , Z ) Sp(4,\mathbb {Z}) the symplectic group. The quotient S p ( 4 , Z ) ∖ h 2 Sp(4,\mathbb {Z})\backslash {h_2} can be interpreted as the moduli space of stable Riemann surfaces of genus 2 2 . This moduli space can be decomposed into two pieces corresponding to the moduli of degenerate and nondegenerate surfaces of genus 2 2 . The decomposition leads to a Mayer-Vietoris sequence in cohomology relating the cohomology of S p ( 4 , Z ) Sp(4,\mathbb {Z}) to the cohomology of the genus two mapping class group Γ 2 0 \Gamma _2^0 . Using this tool, the 3 3 - and 5 5 -primary pieces of the integral cohomology of S p ( 4 , Z ) Sp(4,\mathbb {Z}) are computed.