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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
New series in the Johnson cokernels of the mapping class groups of surfaces

Naoya Enomoto and Takao Satoh

Algebraic & Geometric Topology 14 (2014) 627–669

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

In this paper, we study the Sp–module structure of the cokernel hg,1(k)Im(τk,) of the rational Johnson homomorphism τk, := τk id, where hg,1(k) := hg,1(k) . 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, ) and the Brauer algebras, and our previous work for the Johnson cokernel of the automorphism group of a free group.

Dedicated to the memory of Midori Kato

Johnson homomorphism, mapping class group
Mathematical Subject Classification 2010
Primary: 20G05
Secondary: 57M50
Received: 20 August 2012
Revised: 3 August 2013
Accepted: 27 August 2013
Published: 30 January 2014
Naoya Enomoto
Department of Mathematics
Faculty of Science
Nara Women’s University
Nara city 630-8506
Takao Satoh
Department of Mathematics
Faculty of Science Division II
Tokyo University of Science
Kagurazaka 1-3
Tokyo 1628601