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Homology cylinders: an enlargement of the mapping class group

Jerome Levine

Algebraic & Geometric Topology 1 (2001) 243–270

Erratum: Algebraic & Geometric Topology 1 (2001) 243–270

arXiv: math.GT/0010247

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
Mathematical Subject Classification 2000
Primary: 57N10
Secondary: 57M25
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Publication
Received: 14 November 2000
Accepted: 17 April 2001
Published: 23 April 2001
Authors
Jerome Levine
Department of Mathematics
Brandeis University
Waltham MA 02454-9110
USA
www.math.brandeis.edu/Faculty/jlevine/