Algebraic and Geometric Topology 5 (2005), paper no. 9, pages 135-182.

Algebraic models of Poincare embeddings

Pascal Lambrechts, Don Stanley

Abstract. 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 C<--dT-->P. 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.

Keywords. Poincare embeddings, Lefschetz duality, Sullivan models

AMS subject classification. Primary: 55P62. Secondary: 55M05, 57Q35.

DOI: 10.2140/agt.2005.5.135

E-print: arXiv:math.AT/0503605

Submitted: 8 May 2003. (Revised: 18 August 2004.) Accepted: 21 September 2004. Published: 12 March 2005.

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Pascal Lambrechts, Don Stanley
Institut Mathematique, Universite de Louvain, 2, chemin du Cyclotron
B-1348 Louvain-la-Neuve, Belgium
Department of Mathematics and Statistics, University of Regina
College West 307.14, Regina, Saskatchewan, Canada, S4S 0A2

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