#### 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.

Notes on file formats
Pascal Lambrechts, Don Stanley

Institut Mathematique, Universite de Louvain, 2, chemin du Cyclotron

B-1348 Louvain-la-Neuve, Belgium

and

Department of Mathematics and Statistics, University of Regina

College West 307.14, Regina, Saskatchewan, Canada, S4S 0A2

Email: lambrechts@math.ucl.ac.be, stanley@math.uregina.ca

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