## Algebraic & Geometric Topology

### Clover calculus for homology 3-spheres via basic algebraic topology

#### 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 $(Gn)n$ 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 $(Gn)n$. 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 5, Number 1 (2005), 71-106.

Dates
Accepted: 28 December 2004
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796394

Digital Object Identifier
doi:10.2140/agt.2005.5.71

Mathematical Reviews number (MathSciNet)
MR2135546

Zentralblatt MATH identifier
1072.57008

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds

#### Citation

Auclair, Emmanuel; Lescop, Christine. Clover calculus for homology 3-spheres via basic algebraic topology. Algebr. Geom. Topol. 5 (2005), no. 1, 71--106. doi:10.2140/agt.2005.5.71. https://projecteuclid.org/euclid.agt/1513796394

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