The Birman-Craggs-Johnson homomorphism and abelian cycles in the Torelli group

Tara E. Brendle; Benson Farb

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arXiv · math/0601163

The Birman-Craggs-Johnson homomorphism and abelian cycles in the Torelli group

Tara E. Brendle·Benson Farb

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Abstract

In the 1970s, Birman-Craggs-Johnson used Rochlin's invariant for homology 3-spheres to construct a remarkable surjective homomorphism sigma:I_{g,1}->B_3, where I_{g,1} is the Torelli group and B_3 is a certain F_2-vector space of Boolean (square-free) polynomials. By pulling back cohomology classes and evaluating them on abelian cycles, we construct 16g^4 + O(g^3) dimensions worth of nontrivial elements of H^2(I_{g,1}, F_2) which cannot be detected rationally. These classes in fact restrict to nontrivial classes in the cohomology of the subgroup K_{g,1} < I_{g,1} generated by Dehn twists about separating curves. We also use the ``Casson-Morita algebra'' and Morita's integral lift of the Birman-Craggs-Johnson map restricted to K_{g,1} to give the same lower bound on H^2(K_{g,1},Z).

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Tara E. Brendle, Benson Farb. 2006-01-09. The Birman-Craggs-Johnson homomorphism and abelian cycles in the Torelli group. https://arxiv.org/abs/math/0601163

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