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Characterization of Y₂-Equivalence for Homology Cylinders

Laboratoire Jean Leray, UMR 6629 CNRS/Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 03, France

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and

Jean-Baptiste Meilhan

Laboratoire Jean Leray, UMR 6629 CNRS/Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 03, France

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https://doi.org/10.1142/S0218216503002585Cited by:9 (Source: Crossref)

Abstract

For Σ a compact connected oriented surface, we consider homology cylinders over Σ: these are homology cobordisms with an extra homological triviality condition. When considered up to Y₂-equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian group.

In this paper, when Σ 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.

Keywords:

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