Geometry & Topology

Tree homology and a conjecture of Levine

James Conant, Rob Schneiderman, and Peter Teichner

Full-text: Open access

Abstract

In his study of the group of homology cylinders, J Levine [Algebr. Geom. Topol. 2 (2002) 1197–1204] made the conjecture that a certain group homomorphism η:TD is an isomorphism. Both T and D are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of Out(Fn). In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain T and range D of Levine’s map.

The isomorphism η is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138; arXiv 1202.3463]. In this paper and [arXiv 1202.2482] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders.

Article information

Source
Geom. Topol., Volume 16, Number 1 (2012), 555-600.

Dates
Received: 13 December 2010
Revised: 16 January 2012
Accepted: 16 January 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732389

Digital Object Identifier
doi:10.2140/gt.2012.16.555

Mathematical Reviews number (MathSciNet)
MR2916294

Zentralblatt MATH identifier
1284.57007

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx]

Keywords
Levine conjecture tree homology homology cylinder Whitney tower discrete Morse theory quasi-Lie algebra

Citation

Conant, James; Schneiderman, Rob; Teichner, Peter. Tree homology and a conjecture of Levine. Geom. Topol. 16 (2012), no. 1, 555--600. doi:10.2140/gt.2012.16.555. https://projecteuclid.org/euclid.gt/1513732389


Export citation

References

  • D Cheptea, K Habiro, G Massuyeau, A functorial LMO invariant for Lagrangian cobordisms, Geom. Topol. 12 (2008) 1091–1170
  • T D Cochran, Derivatives of links: Milnor's concordance invariants and Massey's products, Mem. Amer. Math. Soc. 84, no. 427, Amer. Math. Soc. (1990)
  • J Conant, R Schneiderman, P Teichner, Geometric filtrations of string links and homology cylinders
  • J Conant, R Schneiderman, P Teichner, Milnor invariants and twisted Whitney towers
  • J Conant, R Schneiderman, P Teichner, Whitney tower concordance of classical links
  • J Conant, R Schneiderman, P Teichner, Higher-order intersections in low-dimensional topology, Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138
  • J Conant, K Vogtmann, On a theorem of Kontsevich, Algebr. Geom. Topol. 3 (2003) 1167–1224
  • J Conant, K Vogtmann, Morita classes in the homology of automorphism groups of free groups, Geom. Topol. 8 (2004) 1471–1499
  • R Forman, Morse theory for cell complexes, Adv. Math. 134 (1998) 90–145
  • S Garoufalidis, M Goussarov, M Polyak, Calculus of clovers and finite type invariants of $3$–manifolds, Geom. Topol. 5 (2001) 75–108
  • 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”, (M Lyubich, L Takhtajan, editors), Proc. Sympos. Pure Math. 73, Amer. Math. Soc. (2005) 173–203
  • M Goussarov, Finite type invariants and $n$–equivalence of $3$–manifolds, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 517–522
  • N Habegger, Milnor, Johnson and tree-level perturbative invariants, preprint (2000) Available at \setbox0\makeatletter\@url http://www.math.sciences.univ-nantes.fr/~habegger/PS/john100300.ps {\unhbox0
  • N Habegger, G Masbaum, The Kontsevich integral and Milnor's invariants, Topology 39 (2000) 1253–1289
  • N Habegger, W Pitsch, Tree level Lie algebra structures of perturbative invariants, J. Knot Theory Ramifications 12 (2003) 333–345
  • K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1–83
  • K Habiro, G Massuyeau, Symplectic Jacobi diagrams and the Lie algebra of homology cylinders, J. Topol. 2 (2009) 527–569
  • K Habiro, G Massuyeau, From mapping class groups to monoids of homology cobordisms: a survey, preprint (2010) to appear in “Handbook of Teichmüller theory, Vol. III” (A Papadopoulos, editor)
  • D Johnson, A survey of the Torelli group, from: “Low-dimensional topology (San Francisco, Calif., 1981)”, (S J Lomonaco, Jr, editor), Contemp. Math. 20, Amer. Math. Soc. (1983) 165–179
  • M Kontsevich, Formal (non)commutative symplectic geometry, from: “The Gel'fand Mathematical Seminars, 1990–1992”, (L Corwin, I Gel'fand, J Lepowsky, editors), Birkhäuser, Boston (1993) 173–187
  • D N Kozlov, Discrete Morse theory for free chain complexes, C. R. Math. Acad. Sci. Paris 340 (2005) 867–872
  • J Levine, Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001) 243–270
  • J Levine, Addendum and correction to: “Homology cylinders: an enlargement of the mapping class group” [L1?], Algebr. Geom. Topol. 2 (2002) 1197–1204
  • J Levine, Labeled binary planar trees and quasi-Lie algebras, Algebr. Geom. Topol. 6 (2006) 935–948
  • W Magnus, A Karrass, D Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, revised edition, Dover, New York (1976)
  • J Milnor, Link groups, Ann. of Math. 59 (1954) 177–195
  • J Milnor, Isotopy of links. Algebraic geometry and topology, from: “A symposium in honor of S Lefschetz”, (R H Fox, D C D C Spencer, A W Tucker, editors), Princeton Univ. Press (1957) 280–306
  • 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 (Berkeley, CA, 1998)”, (J Hass, M Scharlemann, editors), Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 349–406
  • K E Orr, Homotopy invariants of links, Invent. Math. 95 (1989) 379–394
  • M A Readdy, The pre-WDVV ring of physics and its topology, Ramanujan J. 10 (2005) 269–281
  • C Reutenauer, Free Lie algebras, London Math. Soc. Monogr. (NS) 7, Oxford Science Publ., The Clarendon Press, Oxford Univ. Press, New York (1993)