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Algebraic models of Poincaré embeddings

Pascal Lambrechts and Donald Stanley

Algebraic & Geometric Topology 5 (2005) 135–182

arXiv: math.AT/0503605


Let f : PW 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 := W \ T¯ be its complement. Then W is the homotopy push-out of a diagram C T P. This homotopy push-out square is an example of what is called a Poincaré 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 Poincaré embeddings and we deduce a refinement of the classical Lefschetz duality theorem, giving information on the algebra structure of the cohomology of the complement.

Poincaré embeddings, Lefschetz duality, Sullivan models
Mathematical Subject Classification 2000
Primary: 55P62
Secondary: 55M05, 57Q35
Forward citations
Received: 8 May 2003
Revised: 18 August 2004
Accepted: 21 September 2004
Published: 12 March 2005
Pascal Lambrechts
Institut Mathématique
Université de Louvain
2, chemin du Cyclotron
B-1348 Louvain-la-Neuve
Donald Stanley
Department of Mathematics and Statistics
University of Regina
College West 307.14
Saskatchewan S4S 0A2