Characterization of Y_2-equivalence for homology cylinders

Details Characterization of Y_2-equivalence for homology cylinders Characterization of Y_2-equivalence for homology cylinders

2002-03-18

arxiv.org/abs/math/0203179v2

J. Knot Th. Ramifications 12:4 (2003) 493-522

Mathematics

Geometric Topology

29 pages with 8 figures; a few minor improvements

Scientific paper

For S a compact connected oriented surface, we consider homology cylinders over S: these are homology cobordisms with an extra homological triviality condition. When considered up to Y_2-equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian group. In this paper, when S has one or zero boundary component, we define a surgery map from a certain space of graphs to this group. This map is shown to be an isomorphism, with inverse given by some extensions of the first Johnson homomorphism and Birman-Craggs homomorphisms.

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