Mapping Class Groups of low Genus and their Cohomology

D. J. Benson Mathematical Institute

24–29 St. Giles Oxford OX1 3LB

England

F. R. Cohen Department of Mathematics University of Kentucky Lexington, KY 40506 U.S.A.

Contents

Introduction by D. J. Benson and F. R. Cohen 2

Artin’s Braid Group and the Homology of Certain Subgroups of the Mapping Class Group by F. R. Cohen 7 1 Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 H^{∗}(K_{n}; Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5 The mod-5 Cohomology of Γ_{2,0}^{0} and Γ_{0,0}^{6} . . . . . . . . . . . . . . . . . . . . . . . . 15 6 The mod-3 Cohomology of Γ_{2,0}^{0} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 7 The mod-2 Cohomology of Γ_{2,0}^{0}; Theorems 1.3 and 1.4 . . . . . . . . . . . . . . . . 17 8 H^{∗}(Σ_{6}; H^{∗}(K_{6}; F_{5})); Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 9 Theorem 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 10 Facts about B_{n} and Lemmas 3.2 and 3.3 . . . . . . . . . . . . . . . . . . . . . . . . 25

Specht Modules and the Cohomology of Mapping Class Groups by D. J. Benson 31 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 The modules H^{j}(K_{n}, Z), j ≤ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 Modules for A_{6} and Σ_{6} in characteristic two . . . . . . . . . . . . . . . . . . . . . . 42 4 Diagrams for H^{j}(K_{6}, F_{2}) as F_{2}A_{6}-modules . . . . . . . . . . . . . . . . . . . . . . 49 5 Calculation of H^{∗}(Σ_{6}, H^{∗}(K_{6}, F_{2})) ⇒ H^{∗}(Γ_{0,0}^{6}, F_{2}) . . . . . . . . . . . . . . . . . . 63 6 Finite subgroups of Γ_{0,0}^{6} and Γ_{2,0}^{0} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7 Calculation of the spectral sequences for H^{∗}(Γ_{0,0}^{6}, F_{2}) and H^{∗}(Γ_{2,0}^{0}, F_{2}) . . . . . . 77 8 Calculations in characteristic three . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

The mod 2 cohomology of the mapping class group for a surface of genus two by D. J. Benson and F. R. Cohen 94 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2 A construction for K(Γ_{0,0}^{n}, 1); characteristic classes . . . . . . . . . . . . . . . . . . 96 3 Steenrod operations on H^{∗}(F (S^{2}, 6)/Σ_{6}, F_{2}) . . . . . . . . . . . . . . . . . . . . . . 101 4 Steenrod operations on H^{∗}(Γ_{0,0}^{6}, F_{2}) . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5 The spectral sequence for Γ_{2,0}^{0} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104