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

Benson Farb (University of Chicago); Tara E. Brendle (Louisiana State University)

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

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arxiv math/0601163 v1 pith:OE77WCPY submitted 2006-01-09 math.GT math.GR

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

Tara E. Brendle (Louisiana State University) , Benson Farb (University of Chicago) This is my paper

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keywords birman-craggs-johnsonclassesabeliancohomologyconstructcyclesgrouphomomorphism

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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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The Birman--Craggs--Johnson homomorphism and the handlebody Torelli group
math.GT 2026-07 conditional novelty 6.0

For genus at least 3, the BCJ image of the handlebody Torelli group is an explicit monomial subspace B^bi_3, and of the Johnson kernel is B^bi_2; cup-product lower bounds of order g^6 and g^4 follow.