A homotopy-theoretic proof of Williams's metastable Poincaré embedding theorem
About twenty years ago \textit{B. Williams} proved the following metastable Poincaré embedding theorem geometrically [Compos. Math. 38, 55-66 (1979; Zbl 0402.57029)]: Let \((M,A)\) be a finite, oriented Poincaré pair of dimension \(m\geq 6\), and \(\pi_1(A) \cong \pi_1(M)\). Suppose \(M\) is \(n\)-dimensional, and let \(q=m-n-1\). If \(m\leq 3q\), then any degree \(-1\) map \(\rho: S^m\to M/A\) is induced by a Poincaré embedding \(f:A\to W\). If \(m<3q\), the Poincaré embedding is unique up to concordance. At about the same time Williams also outlined a program for a homotopy theoretic proof of this theorem [Pac. J. Math. 84, 217-224 (1979; Zbl 0523.57008)]. The present paper completes William's program, via the elementary unstable homotopy theory of Barratt, Berstein-Hilton, Boardman-Steer, Ganea, James, and Toda.
scientific article; zbMATH DE number 1042491
scientific article