Algebraic and Geometric Topology 5 (2005),
paper no. 5, pages 71-106.
Clover calculus for homology 3-spheres via basic algebraic topology
Emmanuel Auclair, Christine Lescop
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 (LP-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 LP-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
AMS subject classification.
Primary: 57M27.
Secondary: 57N10.
DOI: 10.2140/agt.2005.5.71
E-print: arXiv:math.GT/0401251
Submitted: 9 February 2004.
Accepted: 28 December 2004.
Published: 3 February 2005.
Notes on file formats
Emmanuel Auclair, Christine Lescop
Institut Fourier (UMR 5582 du CNRS), B.P. 74
38402 Saint-Martin d'Heres cedex, France
Email: auclaire@ujf-grenoble.fr, lescop@ujf-grenoble.fr
URL: http://www-fourier.ujf-grenoble.fr/~lescop
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