We complete the details of a theory outlined by Kontsevich and Soibelman that associates to a semi-algebraic set a certain graded commutative differential algebra of "semi-algebraic differential forms" in a functorial way. This algebra encodes the real homotopy type of the semi-algebraic set in the spirit of the DeRham algebra of differential forms on a smooth manifold. Its development is needed for Kontsevich's proof of the formality of the little cubes operad. P A 6.1. Poincaré Lemma for Ω * P A 6.2. Sheaf propery of Ω * P A 6.3. Extendability of the simplicial set Ω P A (∆ • ) 6.4. The weak equivalence Ω * P A ≃ A P L 7. Monoidal equivalences 8. Oriented semi-algebraic bundles and integration along the fiber 8.1. Properties of SA bundles 8.2. Properties of integration along the fiber

Abstract. We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré duality in the same dimension. This has application in particular to the study of CDGA models of configuration spaces on a closed manifold.

Abstract. We determine the rational homology of the space of long knots in R d for d ≥ 4. Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E 1 page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with a bracket of degree d − 1, which can be obtained as the homology of an explicit graph complex and is in theory completely computable.Our proof is a combination of a relative version of Kontsevich's formality of the little d-disks operad and of Sinha's cosimplicial model for the space of long knots arising from GoodwillieWeiss embedding calculus. As another ingredient in our proof, we introduce a generalization of a construction that associates a cosimplicial object to a multiplicative operad. Along the way we also establish some results about the Bousfield-Kan spectral sequences of a truncated cosimplicial space.

Let M be a smooth manifold and V a Euclidean space. Let Emb(M, V ) be the homotopy fiber of the map Emb(M, V ) −→ Imm(M, V ). This paper is about the rational homology of Emb(M, V ). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M, V ) → HQ ∧ Emb(M, V )+. Our main theorem states that if dim V ≥ 2 ED(M ) + 1 (where ED(M ) is the embedding dimension of M ), the Taylor tower in the sense of orthogonal calculus (henceforward called "the orthogonal tower") of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E 1 . In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad.We write explicit formulas for the layers in the orthogonal tower of the functorThe formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M . This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of Emb(−, V ).

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