A homotopy-theoretic proof of Williams's metastable Poincaré embedding theorem
type: Item
instance of: scholarly article
title: A homotopy-theoretic proof of Williams's metastable Poincaré embedding theorem
zbMATH Open document ID: 0880.55009
DOI: 10.1215/S0012-7094-97-08818-9
author: William Richter
published in: Duke Mathematical Journal
publication date: 1998-02-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57P10
Mathematics Subject Classification ID: 57N35
zbMATH DE Number: 1042491
zbMATH Keywords: metastable embedding theorem
zbMATH Keywords: Poincaré embedding
zbMATH Keywords: unstable homotopy theory
reviewed by: Dušan Repovš
MaRDI profile type: Publication
cites work: Q5517577
cites work: On suspensions and comultiplications
cites work: On Hopf invariants
cites work: Embedding 1-connected manifolds
cites work: Q5586056
cites work: Manifolds with $\pi _1 = Z$
cites work: Poincaré spaces, their normal fibrations and surgery
cites work: Q5649690
cites work: EMBEDDINGS UP TO HOMOTOPY TYPE AND GEOMETRIC SUSPENSIONS OF MANIFOLDS
cites work: A generalization of the homology and homotopy suspension
cites work: Induced Fibrations and Cofibrations
cites work: On the homotopy suspension
cites work: Q4305276
cites work: Q4064722
cites work: Q3675146
cites work: A RELATION BETWEEN POSTNIKOV CLASSES
cites work: On the structure of Poincaré duality spaces
cites work: Equivalence of Toda-Hopf invariants
cites work: Surgery on Poincaré and normal spaces
cites work: The Algebraic Theory of Surgery II. Applications to Topology
cites work: The H-Space Squaring Map on Ω 3 S 4n + 1 Factors through the Double Suspension
cites work: High-Dimensional Knots with π 1 ≅ℤ Are Determined by Their Complements in One More Dimension Than Farber's Range
cites work: Q5522742
cites work: The homotopy category is a homotopy category
cites work: Q5624419
cites work: Q4192669
cites work: Q4187034
cites work: Hopf invariants, localization and embeddings of Poincaré complexes
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description: scientific article; zbMATH DE number 1042491