Geodesic
The Johnson cokernel and the Enomoto–Satoh invariant
Conant, James1
1Department of Mathematics, University of Tennessee, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996, USA
Algebraic and Geometric Topology, Tome 15 (2015) no. 2, pp. 801-821
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We study the cokernel of the Johnson homomorphism for the mapping class group of a surface with one boundary component. A graphical trace map simultaneously generalizing trace maps of Enomoto and Satoh and Conant, Kassabov and Vogtmann is given, and using technology from the author’s work with Kassabov and Vogtmann, this is is shown to detect a large family of representations which vastly generalizes series due to Morita and Enomoto and Satoh. The Enomoto–Satoh trace is the rank-1 part of the new trace, and it is here that the new series of representations is found. The rank-2 part is also investigated, though a fuller investigation of the higher-rank case is deferred to another paper.

DOI : 10.2140/agt.2015.15.801
Classification : 17B40, 20C15, 20F28
Keywords: Johnson homomorphism, Enomoto–Satoh invariant, Johnson cokernel

Conant, James  1

1 Department of Mathematics, University of Tennessee, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996, USA 
@article{10_2140_agt_2015_15_801,
     author = {Conant, James},
     title = {The {Johnson} cokernel and the {Enomoto{\textendash}Satoh} invariant},
     journal = {Algebraic and Geometric Topology},
     pages = {801--821},
     year = {2015},
     volume = {15},
     number = {2},
     doi = {10.2140/agt.2015.15.801},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.801/}
}
TY  - JOUR
AU  - Conant, James
TI  - The Johnson cokernel and the Enomoto–Satoh invariant
JO  - Algebraic and Geometric Topology
PY  - 2015
SP  - 801
EP  - 821
VL  - 15
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.801/
DO  - 10.2140/agt.2015.15.801
ID  - 10_2140_agt_2015_15_801
ER  - 
%0 Journal Article
%A Conant, James
%T The Johnson cokernel and the Enomoto–Satoh invariant
%J Algebraic and Geometric Topology
%D 2015
%P 801-821
%V 15
%N 2
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.801/
%R 10.2140/agt.2015.15.801
%F 10_2140_agt_2015_15_801
Conant, James. The Johnson cokernel and the Enomoto–Satoh invariant. Algebraic and Geometric Topology, Tome 15 (2015) no. 2, pp. 801-821. doi: 10.2140/agt.2015.15.801

[1] M Asada, H Nakamura, On graded quotient modules of mapping class groups of surfaces, Israel J. Math. 90 (1995) 93

[2] J Conant, M Kassabov, Hopf algebras and invariants of the Johnson cokernel, in preparation

[3] J Conant, M Kassabov, K Vogtmann, Higher hairy graph homology, to appear in Geom. Dedicata

[4] J Conant, M Kassabov, K Vogtmann, Hairy graphs and the unstable homology of $\mathrm{Mod}(g,s)$, $\mathrm{Out}(F_n)$ and $\mathrm{Aut}(F_n)$, J. Topol. 6 (2013) 119

[5] H Enomoto, N Enomoto, $\mathrm{Sp}$–irreducible components in the Johnson cokernels of the mapping class groups of surfaces, I, Journal of Lie Theory 24 (2014) 687

[6] N Enomoto, T Satoh, New series in the Johnson cokernels of the mapping class groups of surfaces, Algebr. Geom. Topol. 14 (2014) 627

[7] W Fulton, J Harris, Representation theory, Graduate Texts in Math. 129, Springer (1991)

[8] R Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597

[9] G James, M Liebeck, Representations and characters of groups, Cambridge Univ. Press (1993)

[10] D Johnson, A survey of the Torelli group, from: "Low-dimensional topology" (editor S J Lomonaco Jr.), Contemp. Math. 20, Amer. Math. Soc. (1983) 165

[11] M Larsen, A Shalev, Characters of symmetric groups: Sharp bounds and applications, Invent. Math. 174 (2008) 645

[12] J Levine, Addendum and correction to: “Homology cylinders: An enlargement of the mapping class group” [Algebr. Geom. Topol. 1 (2001), 243–270], Algebr. Geom. Topol. 2 (2002) 1197

[13] S Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993) 699

[14] S Morita, Structure of the mapping class groups of surfaces: A survey and a prospect, from: "Proceedings of the Kirbyfest" (editors J Hass, M Scharlemann), Geom. Topol. Monogr. 2 (1999) 349

[15] S Morita, T Sakasai, M Suzuki, Slides from presentation at Univ. Tokyo (2013)

[16] H Nakamura, Coupling of universal monodromy representations of Galois–Teichmüller modular groups, Math. Ann. 304 (1996) 99

[17] Y Roichman, Upper bound on the characters of the symmetric groups, Invent. Math. 125 (1996) 451

[18] T Satoh, On the lower central series of the IA–automorphism group of a free group, J. Pure Appl. Algebra 216 (2012) 709

Cité par Sources :