The handlebody group and the images of the second Johnson homomorphism

Abstract

The handlebody group and the images of the second Johnson homomorphism QUENTIN FAESGiven an oriented surface bounding a handlebody, we study the subgroup of its mapping class group defined as the intersection of the handlebody group and the second term of the Johnson filtration; A \ J 2 . We introduce two trace-like operators, inspired by Morita's trace, and show that their kernels coincide with the images by the second Johnson homomorphism 2 of J 2 and A \ J 2 , respectively. In particular, we answer in the negative a question asked by Levine about an algebraic description of 2 .A \ J 2 /. By the same techniques, and for a Heegaard surface in S 3 , we also compute the image by 2 of the intersection of the Goeritz group G with J 2 . 57K20 1. Introduction and notation 243 2. Image of the second Johnson homomorphism 2 247 3. Motivation for the study of A \ J 2 256 4. The A-trace 259 5. Computing 2 .A \ J 2 / 272 6. Computing 2 .G \ J 2 / 283 Appendix. Decomposition of 2 .G \ J 2 / ˝Q 287 References 291