Clover calculus for homology 3-spheres via basic algebraic topology

Auclair, Emmanuel; Lescop, Christine

doi:10.2140/agt.2005.5.71

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Clover calculus for homology 3-spheres via basic algebraic topology

Emmanuel Auclair and Christine Lescop

Algebraic & Geometric Topology 5 (2005) 71–106

DOI: 10.2140/agt.2005.5.71

arXiv: math.GT/0401251

Abstract

We present an alternative definition for the Goussarov–Habiro filtration of the $ℤ$ –module freely generated by oriented integral homology 3–spheres, by means of Lagrangian-preserving homology handlebody replacements ( $ℒ P$ –surgeries). Garoufalidis, Goussarov and Polyak proved that the graded space ${(G_{n})}_{n}$ associated to this filtration is generated by Jacobi diagrams. Here, we express elements associated to $ℒ P$ –surgeries as explicit combinations of these Jacobi diagrams in ${(G_{n})}_{n}$ . The obtained coefficient in front of a Jacobi diagram is computed like its weight system with respect to a Lie algebra equipped with a non-degenerate invariant bilinear form, where cup products in 3–manifolds play the role of the Lie bracket and the linking number replaces the invariant form. In particular, this article provides an algebraic version of the graphical clover calculus developed by Garoufalidis, Goussarov, Habiro and Polyak. This version induces splitting formulae for all finite type invariants of homology 3–spheres.

Keywords

3–manifolds, homology spheres, finite type invariants, Jacobi diagrams, Borromeo surgery, clover calculus, clasper calculus, Goussarov–Habiro filtration

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57N10

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Publication

Received: 9 February 2004

Accepted: 28 December 2004

Published: 3 February 2005

Authors

Emmanuel Auclair
Institut Fourier (UMR 5582 du CNRS) BP 74 38402 Saint-Martin d’Hères Cedex France

Christine Lescop
Institut Fourier (UMR 5582 du CNRS) BP 74 38402 Saint-Martin d’Hères Cedex France
http://www-fourier.ujf-grenoble.fr/~lescop/

Volume 5, issue 1 (2005)