Quadratic forms and the Birman-Craggs homomorphisms

Abstract:

Let ${\mathcal {M}_g}$ be the mapping class group of a genus g orientable surface M, and ${\mathcal {J}_g}$ the subgroup of those maps acting trivially on the homology group ${H_1}(M, Z)$. Birman and Craggs produced homomorphisms from ${\mathcal {J}_g}$ to ${Z_2}$ via the Rochlin invariant and raised the question of enumerating them; in this paper we answer their question. It is shown that the homomorphisms are closely related to the quadratic forms on ${H_1}(M, {Z_2})$ which induce the intersection form; in fact, they are in 1-1 correspondence with those quadratic forms of Arf invariant zero. Furthermore, the methods give a description of the quotient of ${\mathcal {J}_g}$ by the intersection of the kernels of all these homomorphisms. It is a ${Z_2}$-vector space isomorphic to a certain space of cubic polynomials over ${H_1}(M, {Z_2})$. The dimension is then computed and found to be $\left ( {\begin {array}{*{20}{c}} {2g} \\ 3 \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} {2g} \\ 2 \\ \end {array} } \right )$ . These results are also extended to the case of a surface with one boundary component, and in this situation the linear relations among the various homomorphisms are also determined.