Volume 14, issue 2 (2014)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 4, 1809–2387
Issue 3, 1225–1808
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
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