Algebraic models of Poincaré 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 $C : = \bar{W \ T}$ be its complement. Then $W$ is the homotopy push-out of a diagram $C \leftarrow \partial T \to P$ . 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.

Keywords

Poincaré embeddings, Lefschetz duality, Sullivan models

Mathematical Subject Classification 2000

Primary: 55P62

Secondary: 55M05, 57Q35

References

Forward citations

Publication

Received: 8 May 2003

Revised: 18 August 2004

Accepted: 21 September 2004

Published: 12 March 2005

Authors

Pascal Lambrechts
Institut Mathématique Université de Louvain 2, chemin du Cyclotron B-1348 Louvain-la-Neuve Belgium

Donald Stanley
Department of Mathematics and Statistics University of Regina College West 307.14 Regina Saskatchewan S4S 0A2 Canada

Volume 5, issue 1 (2005)

Pascal Lambrechts and Donald Stanley

Abstract