Let
be an embedding of a compact polyhedron in a closed oriented manifold
, let
be a regular
neighborhood of
in
and let
be its complement.
Then
is the homotopy
push-out of a diagram
.
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
. When the
codimension is high enough this allows us to completely determine the rational homotopy type of
the complement
.
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.