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

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 \({H^2(\mathcal{K}_{g,1}, {\bf Z})}\) .