A Homotopy-theoretic Proof of Williams’s Metastable Poincaré Embedding Theorem

Recall the notion due to W. Browder of a Poincaré embedding [Br1, Br3, Br2, Br4, Wa, Ra, Wi1, Wi2], which is the homotopy analogue of a smooth embedding of manifolds. Let (M,A) be a simply connected m-dimensional (finite) Poincaré pair. A Poincaré embedding of (M,A) in the sphere S will mean a finite CW-complex W and a map f : A −→ W , such that the homotopy pushout M ∪ι A× [0, 1] ∪f W is homotopy equivalent to S. Given such a Poincaré embedding, B. Williams [Wi1] defines the unstable normal invariant ρ : S −→M/A by collapsing the subspace W . Williams defines two Poincaré embeddings f : A −→W and f ′ : A −→W ′ of (M,A) to be concordant if there exists a homotopy equivalence α : W −→ W ′ so that the maps α · f, f ′ : A −→ W ′ are homotopic. In this paper we give a homotopy theoretic proof of Williams’s metastable Poincaré embedding theorem [Wi1], which was originally proven geometrically.