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

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

2006-01-09

arxiv.org/abs/math/0601163v1

Mathematics

Geometric Topology

21 pages, 7 figures

Scientific paper

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).

Affiliated with

Also associated with

No associations

Rating

The Birman-Craggs-Johnson homomorphism and abelian cycles in the Torelli group does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The Birman-Craggs-Johnson homomorphism and abelian cycles in the Torelli group, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Birman-Craggs-Johnson homomorphism and abelian cycles in the Torelli group will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-683893