Algebraic & Geometric Topology

Clover calculus for homology 3-spheres via basic algebraic topology

Emmanuel Auclair and Christine Lescop

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We present an alternative definition for the Goussarov–Habiro filtration of the –module freely generated by oriented integral homology 3–spheres, by means of Lagrangian-preserving homology handlebody replacements (P–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 P–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

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

Received: 9 February 2004
Accepted: 28 December 2004
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx]

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


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.

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