Witten's conjecture and Property P
type: Item
instance of: scholarly article
title: Witten's conjecture and Property P
zbMATH Open document ID: 1072.57005
published in: Geometry \& Topology
publication date: 2004-10-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0311489
full work available at URL: https://eudml.org/doc/125141
full work available at URL: paper7.abs.html
review text: This important paper is aimed to prove the Property P conjecture, asserting that no homotopy 3-sphere can be obtained from \(S^3\) by Dehn surgery on a non-trivial knot. The conjecture was formulated in the early 60's by \textit{R. H. Bing} [Ann. Math. (2) 77, 210 (1963; Zbl 0115.17304)] and became one of the main open problems in knot theory (Problem 1.15 in Kirby's problem list). In the light of Perelman's recent work, it is worth observing that the property P conjecture actually follows from the Poincaré conjecture [\textit{C. McA. Gordon} and \textit{J. Luecke}, J. Am. Math. Soc. 2, No. 2, 371--415 (1989; Zbl 0678.57005)]. The authors conclude the long history of the conjecture by a masterly combination of different approaches proposed over the years together with recent results from gauge theory and symplectic topology. Denoting by \(K_r\) the 3-manifold resulting from rational \(r\)-surgery on the knot \(K\) in \(S^3\), they prove that \(\pi_1(K_1)\) admits a non-trivial representation in \(\text{SO}(3)\) for any non-trivial knot \(K\). This implies the full conjecture, the case of \(r \neq \pm1\) being already known [\textit{M. Culler}, \textit{C. McA. Gordon}, \textit{J. Luecke} and \textit{P. B. Shalen}, Ann. Math. (2) 125, 237--300 (1987; Zbl 0633.57006)]. Here is a brief outline of the argument. Assume that there exists a non-trivial knot \(K\) which contradicts the thesis. Then, the instanton Floer homology \(HF(K_1)\) is trivial and the Floer exact triangle [\textit{A. Floer}, Hofer, Helmut (ed.) et al., The Floer memorial volume. Birkhäuser Prog. Math. 133, 77--97 (1995; Zbl 0996.57515); \textit{P. J. Braam}, \textit{S. K. Donaldson}, ibid. 195--256 (1995; Zbl 0996.57516)] tells us that also \(HF(K_0)\) is trivial. On the other hand, by Gabai's proof of the Property R conjecture [\textit{D. Gabai}, J. Differ. Geom. 26, 479--536 (1987; Zbl 0639.57008)], \(K_0\) admits a taut foliation by oriented surfaces and it is not homeomorphic to \(S^1 \times S^2\)…
reviewed by: Riccardo Piergallini
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 2105190
zbMATH Keywords: 3-manifold
zbMATH Keywords: knot
zbMATH Keywords: surgery
zbMATH Keywords: homotopy sphere
zbMATH Keywords: gauge theory
author: P. B. Kronheimer
author: T. S. Mrowka
MaRDI profile type: Publication
OpenAlex ID: W3102208073
Wikidata QID: Q55967315
arXiv ID: math/0311489
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DOI: 10.2140/GT.2004.8.295
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description: scientific article