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
(–surgeries).
Garoufalidis, Goussarov and Polyak proved that the graded space
associated
to this filtration is generated by Jacobi diagrams. Here, we express elements associated to
–surgeries
as explicit combinations of these Jacobi diagrams in
.
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.
