Homology cylinders: an enlargement of the mapping class group

By J. Levine. 2000

88 citations recorded by the source corpus.

We consider a homological enlargement of the mapping class group, dened by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov-Habiro theory of nite-type invariants of 3-manifolds. We study the structure of this group from several directions: the relative weight l- tration of Dennis Johnson, the nite-type ltration of Goussarov-Habiro, and the relation to string link concordance. We also consider a new Lagrangian ltration of both the mapping class group and the group of homology cylinders. AMS Classication 57N10; 57M25

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6f1218f754afc468e86fed83829cce9485ef3682

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Citations in the corpus

Bottom tangles and universal invariants (2005, background): “) For each n ≥ 0, the monoid B̄(n, n) contains the Lagrangian submonoid Ln [51] of the monoid of homology cobordisms [17, 21] of a compact, connected, oriented surface Σn,1 of genus 1 and with one boundary component, and hence contains the monoid of homology cylinders (or homologically trivial cobor”
Hairy graphs and the unstable homology of Mod(g, s), Out(Fn) and Aut(Fn) (2011, background): “aturally in several areas of topology and geometric group theory. These include the study of the Johnson ltration of the mapping class group of a surface [17, 23], nite-type invariants of 3-manifolds [4, 10, 20], and the homology of outer automorphism groups of free groups [5, 18, 19]. An important ”
On a theorem of Kontsevich (2002, background): “In the Lie case the Lie algebra LOn is isomorphic to the Lie algebra D∗(Vn)⊗R that has arisen in the study of the mapping class group (see [14], [18]).”
Cohomological structure of the mapping class group and beyond (2005, methodology): “er following them) as well as many interesting questions concerning the structure of Hg,1. It seems that the importance of this group is growing recently. Here we summarize some of the results of [24][60] which will be necessary for our purpose here. Consider Γ = π1(Σg \IntD), which is isomorphic to”