A solution to a conjecture of Zeeman
type: Item
instance of: scholarly article
title: A solution to a conjecture of Zeeman
zbMATH Open document ID: 0729.57009
DOI: 10.1016/0040-9383(91)90028-3
author: Selman Akbulut
published in: Topology
publication date: 1991-01-01T00:00:00Z
review text: Any loop \(\gamma \in S^ 3\) bounds a PL-disk in the 4-ball \(B^ 4\), namely the cone on \(\gamma\). Zeeman conjectured that there exists some contractible 4-manifold W and a loop \(\gamma\subset \partial W\) such that \(\gamma\) cannot bound an imbedded PL-disk in W. In this paper Zeeman's conjecture is proved by constructing such a pair (W,\(\gamma\)). The proof that \(\gamma\) does not bound a PL-disk in W is Theorem 2, from the author's paper ``A fake compact contractible 4-manifold'' [J. Differ. Geom. 33, No.2, 335-356 (1991)]. The main step is the construction of an associated homotopy K3-surface M containing the Brieskorn homology 3- sphere \(\Sigma\) (2,3,7) and a self-diffeomorphism f: \(\partial W\to \partial W\) which extends to a self-homeomorphism of W such that either: (i) M has all its Donaldson polynomial invariants trivial, or (ii) there is no self-diffeomorphism of W extending f. \textit{R. Fintushel} and \textit{R. J. Stern} proved that (i) could not hold [J. Differ. Geom. 34, No.1, 255-265 (1991)], and therefore (ii) must hold, as required for Theorem 2.
Mathematics Subject Classification ID: 57Q35
zbMATH DE Number: 4205411
zbMATH Keywords: loop
zbMATH Keywords: PL-disk
zbMATH Keywords: 4-ball
zbMATH Keywords: 4-manifold
zbMATH Keywords: homotopy K3-surface
zbMATH Keywords: Brieskorn homology 3-sphere
zbMATH Keywords: Donaldson polynomial invariants
zbMATH Keywords: self-diffeomorphism
Wikidata QID: Q123316023
reviewed by: Ian Hambleton
MaRDI profile type: Publication
full work available at URL: 0040-9383(91)90028-3
OpenAlex ID: W2078089591
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description: scientific article; zbMATH DE number 4205411