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

and Farb, B. (2007) The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group. Mathematische Annalen, 338(1), pp. 33-53.

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## Abstract

In the 1970s, Birman–Craggs–Johnson (BCJ) (Trans AMS 237: 283–309, 1978; Trans AMS 261(1):423–422, 1980) used Rochlin’s invariant for homology 3-spheres to construct a remarkable surjective homomorphism $${\sigma:\mathcal{I}_{g,1}\to B_3}$$ , where $${\mathcal{I}_{g,1}}$$ is the Torelli group and B 3 is a certain $${{\bf F}_2}$$ -vector space of Boolean (square-free) polynomials. By pulling back cohomology classes and evaluating them on abelian cycles, we construct $${2g^4 + O(g^3)}$$ dimensions worth of nontrivial elements of $${H^2(\mathcal{I}_{g,1}, {\bf F}_2)}$$ which cannot be detected rationally. These classes in fact restrict to nontrivial classes in the cohomology of the subgroup $${\mathcal{K}_{g,1} < \mathcal{I}_{g,1}}$$ generated by Dehn twists about separating curves. We also use the “Casson–Morita algebra” and Morita’s integral lift of the BCJ map restricted to $${\mathcal{K}_{g,1}}$$ to give the same lower bound on $${H^2(\mathcal{K}_{g,1}, {\bf Z})}$$.

Item Type: Articles Published Yes Brendle, Professor Tara Brendle, T.E., and Farb, B. Q Science > QA Mathematics College of Science and Engineering > School of Mathematics and Statistics > Mathematics Mathematische Annalen 0025-5831 1432-1807

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