Algebraic & Geometric Topology

Homology cylinders: an enlargement of the mapping class group

Jerome Levine

Full-text: Open access

Abstract

We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov–Habiro theory of finite-type invariants of 3–manifolds. We study the structure of this group from several directions: the relative weight filtration of Dennis Johnson, the finite-type filtration of Goussarov–Habiro, and the relation to string link concordance.

We also consider a new Lagrangian filtration of both the mapping class group and the group of homology cylinders.

Article information

Source
Algebr. Geom. Topol., Volume 1, Number 1 (2001), 243-270.

Dates
Received: 14 November 2000
Accepted: 17 April 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882592

Digital Object Identifier
doi:10.2140/agt.2001.1.243

Mathematical Reviews number (MathSciNet)
MR1823501

Zentralblatt MATH identifier
0978.57015

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

Keywords
homology cylinder mapping class group clasper finite-type invariant

Citation

Levine, Jerome. Homology cylinders: an enlargement of the mapping class group. Algebr. Geom. Topol. 1 (2001), no. 1, 243--270. doi:10.2140/agt.2001.1.243. https://projecteuclid.org/euclid.agt/1513882592


Export citation

References

  • J. Birman, R. Craggs, The $\mu$-invariant of $3$-manifolds and certain structural properties of the group of homeomorphisms of a closed oriented $2$-manifold, Transactions of the American Mathematical Society 237 (1978) 283–309.
  • T. Cochran, P. Melvin, Finite type invariants for 3-manifolds, Inventiones Math.140 (1), (2000), 45–100..
  • M. Falk, R. Randell, The lower central series of a fiber-type arrangement, Inventiones Math. 82 (1) (1985), 77–88.
  • S. Garoufalidis, M. Goussarov, M. Polyak, Calculus of clovers and finite type invariants of 3-manifolds, Geometry and Topology 5 (2001) 75–108.
  • S. Garoufalidis, J. Levine, Tree-level invariants of three-manifolds, preprint (1999)..
  • ––––, Finite type 3-manifold invariants, the mapping class group and blinks, J. Diff. Geom. 47 (1997) 257–320.
  • H. B. Griffiths, Automorphisms of a $3$-dimensional handlebody, Abh. Math. Sem. Univ. Hamburg 26(1963) 191-210.
  • M. Goussarov, Finite type invariants and $n$-equivalence of 3-manifolds, C. R. Acad. Sci. Paris Ser. I. Math. 329 (1999) 517–522.
  • N. Habegger, Milnor, Johnson and tree-level perturbative invariants, preprint (2000) www.math.sciences.univ-nantes.fr/~habegger.
  • ––––, X. S. Lin, The classification of Links up to Homotopy, Journal of the A.M.S., 4 (2) (1990) 389–419.
  • K. Habiro, Claspers and finite type invariants of links, Geometry and Topology 4 (2000) 1–83.
  • R. Hain, Infinitesimal presentations of the Torelli groups, Journal of the AMS, 10 (3) (1997) 597–651.
  • D. Johnson, A survey of the Torelli group, Contemporary Math. 20 (1983) 163–179.
  • J. Levine, Pure braids, a new subgroup of the mapping class group and finite-type invariants, Tel Aviv Topology Conference: Rothenberg Festschrift, ed. M. Farber, W. Luck, S. Weinberger, Contemporary Mathematics 231 (1999).
  • E. Luft, Actions of the homeotopy group of an orientable $3$-dimensional handlebody, Math. Annalen 234 (1978) 279–292.
  • S. V. Matveev, Generalized surgery of three-dimensional manifolds and representations of homology spheres, Math. Notices Acad. Sci. USSR, 42:2 (1987) 651–656.
  • S. Morita, Abelian subgroups of the mapping class group of surfaces, Duke Mathematical Jl. 70:3 (1993), 699–726.
  • ––––, Structure of the mapping class group of surfaces: a survey and a prospect, Proceedings of the Kirbyfest, ed. J. Haas and M. Scharlemann, Geometry and Topology monographs, vol. 2 (1999) 349–406.
  • J. Nielsen, Untersuchungen zur Topologie der Geschlössenen Zweiseitigen Flächen I, Acta Mathematica 50 (1927), 189–358.
  • T. Oda, A lower bound for the graded modules associated with the relative weight filtration on the Teichmuller group, preprint.
  • T. Ohtsuki, Finite type invariants of integral homology 3-spheres, J. Knot Theory and its Rami. 5 (1996), 101–115.
  • J. Stallings, Homology and central series of groups, Journal of Algebra 2 (1965), 170–181.