Poincare duality and commutative differential graded algebras

We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincare duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincare duality in the same dimension. This has application in particular to the study of CDGA models of configuration spaces on a closed manifold.Comment: 14 pages. Final version to be published in Annales Scientifiques EN

A remarkable DG-module model for configuration spaces

Let M be a simply-connected closed manifold and consider the (ordered) configuration space of $k$ points in M, F(M,k). 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.Comment: Minor revisio

Calculus of functors, operad formality, and rational homology of embedding spaces

Let M be a smooth manifold and V a Euclidean space. Let Ebar(M,V) be the homotopy fiber of the map from Emb(M,V) to Imm(M,V). This paper is about the rational homology of Ebar(M,V). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M,V) |--> HQ /\Ebar(M,V)_+. Our main theorem states that if the dimension of V is more than twice 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 functor HQ /\Ebar(M,V)_+. The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homology type of M. This, together with our rational splitting theorem, implies that under the above assumption on codimension, the rational homology groups of Ebar(M,V) are determined by the rational homology type of M.Comment: 35 pages. An erroneous definition in the last section was corrected, as well as several misprints. The introduction was somewhat reworked. The paper was accepted for publication in Acta Mathematic

Configuration Spaces of Manifolds with Boundary

We study ordered configuration spaces of compact manifolds with boundary. We show that for a large class of such manifolds, the real homotopy type of the configuration spaces only depends on the real homotopy type of the pair consisting of the manifold and its boundary. We moreover describe explicit real models of these configuration spaces using three different approaches. We do this by adapting previous constructions for configuration spaces of closed manifolds which relied on Kontsevich's proof of the formality of the little disks operads. We also prove that our models are compatible with the richer structure of configuration spaces, respectively a module over the Swiss-Cheese operad, a module over the associative algebra of configurations in a collar around the boundary of the manifold, and a module over the little disks operad.Comment: 107 page

Algebraic models of Poincare embeddings

Let f: P-->W be an embedding of a compact polyhedron in a closed oriented manifold W, let T be a regular neighborhood of P in W and let C:=closure(W-T) be its complement. Then W is the homotopy push-out of a diagram CP. This homotopy push-out square is an example of what is called a Poincare embedding. We study how to construct algebraic models, in particular in the sense of Sullivan, of that homotopy push-out from a model of the map f. When the codimension is high enough this allows us to completely determine the rational homotopy type of the complement C = W-f(P). Moreover we construct examples to show that our restriction on the codimension is sharp. Without restriction on the codimension we also give differentiable modules models of Poincare embeddings and we deduce a refinement of the classical Lefschetz duality theorem, giving information on the algebra structure of the cohomology of the complement.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-9.abs.htm