Algebraic and Geometric Topology 1 (2001), paper no. 12, pages 243-270.

Homology cylinders: an enlargement of the mapping class group

Jerome Levine

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.

Keywords. Homology cylinder, mapping class group, clasper, finite-type invariant

AMS subject classification. Primary: 57N10. Secondary: 57M25.

DOI: 10.2140/agt.2001.1.243

E-print: arXiv:math.GT/0010247

Submitted: 14 November 2000. Accepted: 17 April 2001. Published: 23 April 2001.

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Jerome Levine
Department of Mathematics, Brandeis University
Waltham, MA 02454-9110, USA
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