Nathan Habegger

The Kontsevich integral and Milnor’s invariants

Abstract A formula for computing the Milnor (concordance) invariants from the Kontsevich integral is obtained. The reduced Kontsevich integral (with values in the quotient by all loop diagrams) is shown to be the universal concordance invariant of finite type. Some applications are discussed.

Une variété de dimension 4 avec forme d'intersection paire et signature-8

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Knots and links in codimension greater than 2

Abstract A classification of links in codimension greater than 2 in the manifold S 1 × R n − 1 is obtained. Exact sequences involving classical knot and link groups in codimension greater than 2 are given a surgery theoretical interpretation.

Tree level Lie algebra structures of perturbative invariants

We study two different Lie algebra structures on the space of Feynman diagrams at tree level. We show that each such structure arises naturally from a tower of automorphism groups of nilpotent quotients of a free group.

The topological IHX relation, pure braids, and the Torelli group

We prove that the filtration on the pure braid group on g strands, induced by the lower central series of the Torelli group of a genus g surface with one boundary component, coincides with its lower central series, shifted by one. In particular, the cubic Jacobi relations in the pure braid group are quadratic relations in the Torelli group.

THE TOPOLOGICAL IHX RELATION

We give an exposition of the classification of finite type invariants of homology 3-spheres. A new conceptually-based proof of the topological IHX relation, needed to show the well-definedness of diagrams to manifolds, is given

Obstructions to embedding disks II (a proof of a conjecture of Hudson)

Abstract For a manifold W of dimension m with boundary ∂W , the vanishing of certain intersection and self-intersection invariants is a necessary condition to embedding n -disks. For ( W , ∂W ) 2 n − m connected, it is also sufficient.

Embedding up to homotopy type— The first obstruction

Abstract For a 2 n − m connected map from an n -dimensional complex to a m -dimensional manifold, an obstruction to embedding up to homotopy type is defined. The vanishing of this obstruction is a necessary and sufficient condition (in the 2 n − m connected case, 2 n − m ⩾ 2, m − n ⩾3) to obtain an embedding up to homotopy type. In case the target manifold is Euclidean space, it is shown that the obstruction vanishes if and only if certain Thom operations are trivial. A classification theorem is given in the 2 n − m +1 connected case.

ON THE CLASSIFICATION OF LINKS UP TO FINITE TYPE

We use an action, of 2l-component string links on l-component string links, defined by the first author and Xiao-Song Lin, to lift the indeterminacy of finite type link invariants. The set of links up to this new indeterminacy is in bijection with the orbit space of the restriction of this action to the stabilizer of the identity. Structure theorems for the sets of links up to C_n-equivalence and Self-C_n-equivalence are also given.

Homotopy classes of 2 disjoint 2p-spheres in R3p + 1

Abstract For p ≠ 3,7, p > 1, link homotopy classes of link maps S 2p ⊔ S 2p → S 3p + 1 are classified using the α-invariant. It is shown that any such link map is link homotopic to an embedded link. These results are obtained through a study of the homotopy type of the complement of the image of a map from one sphere to another.