Homology cylinders: an enlargement of the mapping class group

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.

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-12.abs.html

References in corpus (4)

• Claspers and finite type invariants of links
• Calculus of clovers and finite type invariants of 3-manifolds
• Finite type invariants of 3-manifolds
• Pure braids, a new subgroup of the mapping class group and finite type invariants

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