A relative Euclidean thickening

Abstract

For a finite CW pair $(K,L)$, we show how to construct a Poincaré triad $(P;{\partial }_{0}P,{\partial }_{1}P)$ and a weak homotopy equivalence $(K,L)\stackrel{{}_{\sim }}{\to }(P,{\partial }_{0}P)$. Furthermore, the Poincaré pair $(P,\partial P)$ has a trivial Spivak normal fibration. The proof is homotopy-theoretic. We also prove a uniqueness result.

Added Addendum C which refines the uniqueness statement. Fixed more typos