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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quadratic forms and the Birman-Craggs homomorphisms

Author: Dennis Johnson
Journal: Trans. Amer. Math. Soc. 261 (1980), 235-254
MSC: Primary 57N10; Secondary 57N05
MathSciNet review: 576873
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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.

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Article copyright: © Copyright 1980 American Mathematical Society

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