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

Brendle, T.E. and Farb, B. (2007) The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group. Mathematische Annalen, 338(1), pp. 33-53. (doi: 10.1007/s00208-006-0066-y)

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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
Glasgow Author(s) Enlighten ID:Brendle, Professor Tara
Authors: Brendle, T.E., and Farb, B.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Mathematische Annalen
ISSN (Online):1432-1807

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