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

Enomoto, Naoya¹ ; Satoh, Takao²

¹Department of Mathematics, Faculty of Science, Nara Women’s University, Kitauoyahigashi-machi, Nara city 630-8506, Japan
²Department of Mathematics, Faculty of Science Division II, Tokyo University of Science, Kagurazaka 1-3, Shinjuku-ku, Tokyo 1628601, Japan

Algebraic and Geometric Topology, Tome 14 (2014) no. 2, pp. 627-669

Résumé

Let Σg,1 be a compact oriented surface of genus g with one boundary component, and ℳg,1 its mapping class group. Morita showed that the image of the kth Johnson homomorphism τkℳ of ℳg,1 is contained in the kernel hg,1(k) of an Sp–equivariant surjective homomorphism H ⊗ℤℒ2g(k + 1) →ℒ2g(k + 2), where H := H1(Σg,1, ℤ) and ℒ2g(k) is the degree k part of the free Lie algebra ℒ2g generated by H.

In this paper, we study the Sp–module structure of the cokernel hg,1ℚ(k)∕Im(τk,ℚℳ) of the rational Johnson homomorphism τk,ℚℳ := τkℳ⊗ idℚ, where hg,1ℚ(k) := hg,1(k) ⊗ℤℚ. In particular, we show that the irreducible Sp–module corresponding to a partition [1k] appears in the kth Johnson cokernel for any k ≡ 1(mod4) 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 [1k] 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.

Détail

DOI : 10.2140/agt.2014.14.627

Classification : 20G05, 57M50
Keywords: Johnson homomorphism, mapping class group

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