New series in the Johnson cokernels of the mapping class groups of surfaces

Abstract

Let Σ g,1 be a compact oriented surface of genus g with one boundary component, and M g,1 its mapping class group. Morita showed that the image of the k-th Johnson homomorphism τ M k of M g,1 is contained in the kernel h g,1 (k) of an Sp-equivariant surjective homomorphism H ⊗ Z L 2g (k + 1) → L 2g (k + 2), where H := H 1 (Σ g,1 , Z) and L 2g (k) is the degree k-part of the free Lie algebra L 2g generated by H.In this paper, we study the Sp-module structure of the cokernelIn particular, we show that the irreducible Sp-module corresponding to a partition [1 k ] appears in the k-th Johnson cokernel for any k ≡ 1 (mod 4) and k ≥ 5 with multiplicity one. We also give a new proof of the fact due to Morita that the irreducible Sp-module corresponding to a partition [k] appears in the Johnson cokernel with multiplicity one for odd k ≥ 3.The strategy of the paper is to give explicit descriptions of maximal vectors with highest weight [1 k ] and [k] in the Johnson cokernel. Our construction is inspired by the Brauer-Schur-Weyl duality between Sp(2g, Q) and the Brauer algebras, and our previous work for the Johnson cokernel of the automorphism group of a free group.

A survey of the Johnson homomorphisms of the automorphism groups of free groups and related topics

“…Combining Hain's result above and the fact that Coker(τ We wrote down a maximal vector of [1 k ] in [17]. In [49], Nakamura and Tsunogai gave a table of the Sp-irreducible decompositions of h Q g,1 (k) for 1 ≤ k ≤ 15, which are calculated by a computer.…”

Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra

“…The Lie algebra a g appeared in a recent work of Enomoto and Satoh [6] and also in Kawazumi and Kuno [13], where they found certain new roles of this Lie algebra. We refer to the above cited papers for details.…”

Johnson-Morita theory in mapping class groups and monoids of homology cobordisms of surfaces

“…Generalizing the trace maps, Enomoto and Satoh [23] found more components in the cokernel of τ Q . Different approaches to the cokernel were given by Conant-Kassabov-Vogtmann [17] and Kawazumi-Kuno [53].…”