Christine Lescop

Clover calculus for homology 3-spheres via basic algebraic topology

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 re- placements (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 co- efficient 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 devel- oped by Garoufalidis, Goussarov, Habiro and Polyak. This version induces splitting formulae for all finite type invariants of homology 3-spheres. AMS Classification 57M27; 57N10

ABOUT THE UNIQUENESS OF THE KONTSEVICH INTEGRAL

We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal Vassiliev link invariants: the Kontsevich Integral and the perturbative expression of the Chern-Simons theory. As a corollary, we prove that the Altschuler and Freidel anomaly -that groups the Bott and Taubes anomalous terms- is a combination of diagrams with two univalent vertices and we explicitly define the isomorphism of which transforms the Kontsevich integral into the Poirier limit of the perturbative expression of the Chern-Simons theory for framed links, as a function of α

On the kernel of the Casson invariant

Abstract We prove: Any integral homology sphere with Casson invariant zero can be obtained from S3 by surgery on a boundary link each component of which has a trivial Alexander polynomial.

Surgery formulae for finite type invariants of rational homology 3–spheres

We first present four graphic surgery formulae for the degree n part Zn of the KontsevichKuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these four formulae determines an alternate sum of the form X I½N (i1) ]I Zn(MI) where N is a finite set of disjoint operations to be performed on a rational homology sphere M, and MI denotes the manifold resulting from the operations in I. The first formula treats the case when N is a set of 2n Lagrangian-preserving surgeries (a Lagrangian-preserving surgery replaces a rational homology handlebody by another such without changing the linking numbers of curves in its exterior). In the second formula, N is a set of n Dehn surgeries on the components of a boundary link. The third formula deals with the case of 3n surgeries on the components of an algebraically split link. The fourth formula is for 2n surgeries on the components of an algebraically split link in which all Milnor triple linking numbers vanish. In the case of homology spheres, these formulae can be seen as a refinement of the Garoufalidis-Goussarov-Polyak comparison of different filtrations of the rational vector space freely generated by oriented homology spheres (up to orientationpreserving homeomorphisms). The presented formulae are then applied to the study of the variation of Zn under a p/qsurgery on a knot K. This variation is a degree n polynomial in q/p when the class of q/p in Q/Z is fixed, and the coefficients of these polynomials are knot invariants, f or which various topological properties or topological definitions are given.

On Homotopy Invariants of Combings of Three-manifolds

Combings of oriented compact 3-manifolds are homotopy classes of nowhere zero vector fields in these manifolds. A first known invariant of a combing is its Euler class, that is the Euler class of the normal bundle to a combing representative in the tangent bundle of the 3-manifold M . It only depends on the Spinc-structure represented by the combing. When this Euler class is a torsion element of H2(M ;Z), we say that the combing is a torsion combing. Gompf introduced a Q-valued invariant θG of torsion combings of closed 3-manifolds that distinguishes all combings that represent a given Spinc-structure. This invariant provides a grading of the Heegaard Floer homology ĤF for manifolds equipped with torsion Spinc-structures. We give an alternative definition of the Gompf invariant and we express its variation as a linking number. We also define a similar invariant p1 for combings of manifolds bounded by S2. We show that the Θ-invariant, that is the simplest configuration space integral invariant of rational homology spheres, is naturally an invariant of combings of rational homology balls, that reads (14p1 + 6λ) where λ is the Casson-Walker invariant. The article also includes a mostly self-contained presentation of combings.