## Algebraic & Geometric Topology

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

#### Abstract

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 2 (2014), 627-669.

Dates
Revised: 3 August 2013
Accepted: 27 August 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715842

Digital Object Identifier
doi:10.2140/agt.2014.14.627

Mathematical Reviews number (MathSciNet)
MR3159965

Zentralblatt MATH identifier
1347.20049

Subjects
Primary: 20G05: Representation theory
Secondary: 57M50: Geometric structures on low-dimensional manifolds

#### Citation

Enomoto, Naoya; Satoh, Takao. New series in the Johnson cokernels of the mapping class groups of surfaces. Algebr. Geom. Topol. 14 (2014), no. 2, 627--669. doi:10.2140/agt.2014.14.627. https://projecteuclid.org/euclid.agt/1513715842

#### References

• S Andreadakis, On the automorphisms of free groups and free nilpotent groups, Proc. London Math. Soc. 15 (1965) 239–268
• M Asada, H Nakamura, On graded quotient modules of mapping class groups of surfaces, Israel J. Math. 90 (1995) 93–113
• N Bourbaki, Lie groups and Lie algebras, Chapters 1–3, Elements of Mathematics, Springer, Berlin (1989)
• J Conant, M Kassabov, K Vogtmann, Hairy graphs and the unstable homology of ${\rm Mod}(g,s)$, ${\rm Out}(F\sb n)$ and ${\rm Aut}(F\sb n)$, J. Topol. 6 (2013) 119–153
• N Enomoto, T Satoh, On the derivation algebra of the free Lie algebra and trace maps, Algebr. Geom. Topol. 11 (2011) 2861–2901
• W Fulton, J Harris, Representation theory: A first course, Graduate Texts in Mathematics 129, Springer, New York (1991)
• A M Garsia, Combinatorics of the free Lie algebra and the symmetric group, from: “Analysis, et cetera”, (P H Rabinowitz, E Zehnder, editors), Academic Press, Boston, MA (1990) 309–382
• R Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597–651
• J Hu, Dual partially harmonic tensors and Brauer–Schur–Weyl duality, Transform. Groups 15 (2010) 333–370
• J Hu, Y Yang, Some irreducible representations of Brauer's centralizer algebras, Glasg. Math. J. 46 (2004) 499–513
• D Johnson, An abelian quotient of the mapping class group ${\cal I}\sb{g}$, Math. Ann. 249 (1980) 225–242
• D Johnson, The structure of the Torelli group, I: A finite set of generators for ${\cal I}$, Ann. of Math. 118 (1983) 423–442
• D Johnson, The structure of the Torelli group, II: A characterization of the group generated by twists on bounding curves, Topology 24 (1985) 113–126
• D Johnson, The structure of the Torelli group, III: The abelianization of $\mathcal{T}$, Topology 24 (1985) 127–144
• A A Kljačko, Lie elements in a tensor algebra, Sibirsk. Mat. Ž. 15 (1974) 1296–1304, 1430
• K Koike, I Terada, Young-diagrammatic methods for the representation theory of the classical groups of type $B\sb n,\;C\sb n,\;D\sb n$, J. Algebra 107 (1987) 466–511
• M Kontsevich, Formal (non)commutative symplectic geometry, from: “The Gel'fand Mathematical Seminars, 1990–1992”, (L Corwin, I Gelrime fand, J Lepowsky, editors), Birkhäuser, Boston, MA (1993) 173–187
• M Kontsevich, Feynman diagrams and low-dimensional topology, from: “First European Congress of Mathematics, Vol. II”, (A Joseph, F Mignot, F Murat, B Prum, R Rentschler, editors), Progr. Math. 120, Birkhäuser, Basel (1994) 97–121
• W Kraśkiewicz, J Weyman, Algebra of coinvariants and the action of a Coxeter element, Bayreuth. Math. Schr. (2001) 265–284
• I G Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford Mathematical Monographs, The Clarendon Press, New York (1995)
• S Morita, Casson's invariant for homology $3$–spheres and characteristic classes of surface bundles, I, Topology 28 (1989) 305–323
• S Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993) 699–726
• S Morita, Structure of the mapping class groups of surfaces: a survey and a prospect, from: “Proceedings of the Kirbyfest”, (J Hass, M Scharlemann, editors), Geom. Topol. Monogr. 2 (1999) 349–406
• H Nakamura, H Tsunogai, Atlas of pro-$l$ mapping class groups and related topics, in preparation
• J Nielsen, Die Isomorphismengruppe der freien Gruppen, Math. Ann. 91 (1924) 169–209
• A Ram, Characters of Brauer's centralizer algebras, Pacific J. Math. 169 (1995) 173–200
• C Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series 7, The Clarendon Press, New York (1993)
• T Satoh, On the lower central series of the IA–automorphism group of a free group, J. Pure Appl. Algebra 216 (2012) 709–717
• J-P Serre, Lie algebras and Lie groups, 2nd edition, Lecture Notes in Mathematics 1500, Springer, Berlin (1992)
• E Witt, Faithful representation of Lie rings (Treue Darstellung Liescher Ringe), J. Reine Angew. Math. 177 (1937) 152–160