Pascal Lambrechts

Formality of the little N-disks operad

Introduction Notation, linear orders, weak partitions, and operads CDGA models for operads Real homotopy theory of semi-algebraic sets The Fulton-MacPherson operad The CDGAs of admissible diagrams Cooperad structure on the spaces of (admissible) diagrams Equivalence of the cooperads D and H * (C[ * ]) The Kontsevich configuration space integrals Proofs of the formality theorems Index of notation Bibliography

The rational homology of spaces of long knots in codimension > 2

We determine the rational homology of the space of long knots in R for d 4 . Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with bracket of degree d 1 , which can be obtained as the homology of an explicit graph complex and is in theory completely computable.

Homotopy graph-complex for configuration and knot spaces

We prove that the primitive part of the Sinha homology spectral sequence E2-term for the space of long knots is rationally isomorphic to the homotopy epsilon 2-term. We also define natural graph-complexes computing the rational homotopy of configuration and of knot spaces.

Embeddings up to homotopy of two-cones in euclidean space

We say that a finite CW-complex X embeds up to homotopy in a sphere S n+1 if there exists a subpolyhedron K ⊂ S n+1 having the homotopy type of X. The main result of this paper is a sufficient condition for the existence of such a homotopy embedding in a given codimension when X is a simply-connected two-cone (a two-cone is the homotopy cofibre of a map between two suspensions). We give different applications of this result: we prove that if X is a two-cone then there are no rational obstructions to embeddings up to homotopy in codimension 3. We give also a description of the homotopy type of the boundary of a regular neighborhood of the embedding of a two-cone in a sphere. This enables us to construct a closed manifold M whose Lusternik-Schnirelmann category and cone-length are not affected by removing one point of M.

A remarkable DGmodule model for configuration spaces

Let M be a simply connected closed manifold and consider the (ordered) configuration space F(M, k) of k points in M. In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the rational homotopy type of F(M, k). We prove that our model it is at least a Sigma(k)-equivariant differential graded model. We also study Lefschetz duality at the level of cochains and describe equivariant models of the complement of a union of polyhedra in a closed manifold.

Examples of rational homotopy types of blow-ups

We give an example of two homotopic embeddings j(0), j(1) : V -> W of manifolds with isomorphic complex normal bundles but such that the blow-ups of W along j(0) and along j(1) have different rational homotopy types.