Algebraic & Geometric Topology

Homology cylinders: an enlargement of the mapping class group

Jerome Levine

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

homology cylinder mapping class group clasper finite-type invariant


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.

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