Algebraic models of Poincaré embeddings

Abstract

Let $f:P\hookrightarrow W$ 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 $C:=\overline{W\backslash T}$ 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 $C\simeq W\backslash f\left(P\right)$. 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.