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.
Auclair, Emmanuel 1 ; Lescop, Christine 1
@article{10_2140_agt_2005_5_71,
author = {Auclair, Emmanuel and Lescop, Christine},
title = {Clover calculus for homology 3-spheres via basic algebraic topology},
journal = {Algebraic and Geometric Topology},
pages = {71--106},
year = {2005},
volume = {5},
number = {1},
doi = {10.2140/agt.2005.5.71},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.71/}
}
TY - JOUR AU - Auclair, Emmanuel AU - Lescop, Christine TI - Clover calculus for homology 3-spheres via basic algebraic topology JO - Algebraic and Geometric Topology PY - 2005 SP - 71 EP - 106 VL - 5 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.71/ DO - 10.2140/agt.2005.5.71 ID - 10_2140_agt_2005_5_71 ER -
%0 Journal Article %A Auclair, Emmanuel %A Lescop, Christine %T Clover calculus for homology 3-spheres via basic algebraic topology %J Algebraic and Geometric Topology %D 2005 %P 71-106 %V 5 %N 1 %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.71/ %R 10.2140/agt.2005.5.71 %F 10_2140_agt_2005_5_71
Auclair, Emmanuel; Lescop, Christine. Clover calculus for homology 3-spheres via basic algebraic topology. Algebraic and Geometric Topology, Tome 5 (2005) no. 1, pp. 71-106. doi: 10.2140/agt.2005.5.71
[1] , , , Calculus of clovers and finite type invariants of 3–manifolds, Geom. Topol. 5 (2001) 75
[2] , , Notes sur l’invariant de Casson des sphères d’homologie de dimension trois, Enseign. Math. (2) 38 (1992) 233
[3] , Milnor, Johnson and tree level perturbative invariants, preprint (2000)
[4] , Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1
[5] , , Perturbative 3–manifold invariants by cut-and-paste topology
[6] , An invariant of integral homology 3–spheres which is universal for all finite type invariants, from: "Solitons, geometry, and topology: on the crossroad", Amer. Math. Soc. Transl. Ser. 2 179, Amer. Math. Soc. (1997) 75
[7] , A sum formula for the Casson–Walker invariant, Invent. Math. 133 (1998) 613
[8] , Splitting formulae for the Kontsevich–Kuperberg–Thurston invariant of rational homology 3–spheres
[9] , An introduction to knot theory, 175, Springer (1997)
[10] , , , Combinatorial group theory, Dover Publications (1976)
[11] , Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987) 268, 345
[12] , , On a certain move generating link-homology, Math. Ann. 284 (1989) 75
[13] , Finite type invariants of integral homology 3–spheres, J. Knot Theory Ramifications 5 (1996) 101
[14] , Knots and links, 7, Publish or Perish (1976)
[15] , The geometry and topology of 3–manifolds, lecture notes, Princeton University (1978)
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