Abelian quotients of the Y –filtration on the homology cylinders via the LMO functor

Nozaki, Yuta¹ ; Sato, Masatoshi² ; Suzuki, Masaaki³

¹ Organization for the Strategic Coordination of Research and Intellectual Properties, Meiji University, Tokyo, Japan, Graduate School of Advanced Science and Engineering, Hiroshima University, Hiroshima, Japan
² Department of Mathematics, Tokyo Denki University, Tokyo, Japan
³ Department of Frontier Media Science, Meiji University, Tokyo, Japan

Geometry & topology, Tome 26 (2022) no. 1, pp. 221-282.

Résumé

We construct a series of homomorphisms from the $Y$–filtration on the monoid of homology cylinders to torsion modules via the mod $\mathbb{Z}$ reduction of the LMO functor. The restrictions of our homomorphisms to the lower central series of the Torelli group do not factor through Morita’s refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. We also determine the third graded quotient $Y_3\mathcal{ℐ}{\mathcal{𝒞}}_{g,1}∕Y_4$ of the $Y$–filtration.

Détail

Keywords: Torelli group, Johnson kernel, homology cylinder, LMO functor, clasper, Jacobi diagram, Johnson homomorphism, Sato–Levine invariant

Bibliographie

[1] B Bollobás, Graph theory: an introductory course, 63, Springer (1979) | DOI

[2] D Cheptea, K Habiro, G Massuyeau, A functorial LMO invariant for Lagrangian cobordisms, Geom. Topol. 12 (2008) 1091 | DOI

[3] S Chmutov, S Duzhin, J Mostovoy, Introduction to Vassiliev knot invariants, Cambridge Univ. Press (2012) | DOI

[4] J Conant, R Schneiderman, P Teichner, Tree homology and a conjecture of Levine, Geom. Topol. 16 (2012) 555 | DOI

[5] J Conant, R Schneiderman, P Teichner, Whitney tower concordance of classical links, Geom. Topol. 16 (2012) 1419 | DOI

[6] J Conant, R Schneiderman, P Teichner, Geometric filtrations of string links and homology cylinders, Quantum Topol. 7 (2016) 281 | DOI

[7] S Garoufalidis, M Goussarov, M Polyak, Calculus of clovers and finite type invariants of 3–manifolds, Geom. Topol. 5 (2001) 75 | DOI

[8] S Garoufalidis, J Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, from: "Graphs and patterns in mathematics and theoretical physics" (editors M Lyubich, L Takhtajan), Proc. Sympos. Pure Math. 73, Amer. Math. Soc. (2005) 173 | DOI

[9] M Goussarov, Finite type invariants and n–equivalence of 3–manifolds, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 517 | DOI

[10] M N Gusarov, Variations of knotted graphs : the geometric technique of n–equivalence, Algebra i Analiz 12 (2000) 79

[11] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1 | DOI

[12] K Habiro, G Massuyeau, Symplectic Jacobi diagrams and the Lie algebra of homology cylinders, J. Topol. 2 (2009) 527 | DOI

[13] K Habiro, G Massuyeau, From mapping class groups to monoids of homology cobordisms: a survey, from: "Handbook of Teichmüller theory, III" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 17, Eur. Math. Soc. (2012) 465 | DOI

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

[15] A Heap, Bordism invariants of the mapping class group, Topology 45 (2006) 851 | DOI

[16] D Johnson, The structure of the Torelli group, III : The abelianization of T , Topology 24 (1985) 127 | DOI

[17] J Levine, Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001) 243 | DOI

[18] J Levine, Addendum and correction to “Homology cylinders : an enlargement of the mapping class group”, Algebr. Geom. Topol. 2 (2002) 1197 | DOI

[19] J Levine, Labeled binary planar trees and quasi-Lie algebras, Algebr. Geom. Topol. 6 (2006) 935 | DOI

[20] G Massuyeau, Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant, Bull. Soc. Math. France 140 (2012) 101 | DOI

[21] G Massuyeau, J B Meilhan, Characterization of Y 2–equivalence for homology cylinders, J. Knot Theory Ramifications 12 (2003) 493 | DOI

[22] G Massuyeau, J B Meilhan, Equivalence relations for homology cylinders and the core of the Casson invariant, Trans. Amer. Math. Soc. 365 (2013) 5431 | DOI

[23] J B Meilhan, On surgery along Brunnian links in 3–manifolds, Algebr. Geom. Topol. 6 (2006) 2417 | DOI

[24] S Morita, Casson’s invariant for homology 3–spheres and characteristic classes of surface bundles, I, Topology 28 (1989) 305 | DOI

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

[26] 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, Geom. Topol. Publ. (1999) 349 | DOI

[27] S Morita, T Sakasai, M Suzuki, Torelli group, Johnson kernel, and invariants of homology spheres, Quantum Topol. 11 (2020) 379 | DOI

[28] T Ohtsuki, Quantum invariants : a study of knots, 3–manifolds, and their sets, 29, World Sci. (2002)

[29] J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170 | DOI