A surgery proof of Bing’s theorem characterizing the 3-sphere. (English) Zbl 1051.57015
A well-known characterization of the 3-sphere by R. H. Bing [Ann. Math. (2) 68, 17–37 (1958; Zbl 0081.39202)] asserts that a closed connected 3-manifold \(M\) is homeomorphic to the 3-sphere if and only if every knot in \(M\) is contained in a 3-ball. Unfortunately Bing’s proof contained a gap [cf. ibid. 77, 210 (1963; Zbl 0115.17304)]. Namely, the property P conjecture was still missing at that time (recently, P. B. Kronheimer and T. S. Mrowka proved such a conjecture in [Geom. Topol. 8, 295–310 (2004; Zbl 1072.57005)]).
Nowadays, alternative proofs of Bing’s theorem can be found, also in textbooks: J. Hempel [Ann. Math. Studies 86 (1976; Zbl 0345.57001)]; D. Rolfsen [Math. Lecture Series 7, 2nd print with corr., Publish or Perish (1990; Zbl 0854.57002)]; R. Myers [Proc. Am. Math. Soc. 72, 397–402 (1978; Zbl 0395.57002)].
In this short paper, the author gives another proof of Bing’s characterization of the 3-sphere, based on an elementary surgery argument.
Nowadays, alternative proofs of Bing’s theorem can be found, also in textbooks: J. Hempel [Ann. Math. Studies 86 (1976; Zbl 0345.57001)]; D. Rolfsen [Math. Lecture Series 7, 2nd print with corr., Publish or Perish (1990; Zbl 0854.57002)]; R. Myers [Proc. Am. Math. Soc. 72, 397–402 (1978; Zbl 0395.57002)].
In this short paper, the author gives another proof of Bing’s characterization of the 3-sphere, based on an elementary surgery argument.
Reviewer: Riccardo Piergallini (Camerino)
MSC:
| 57M40 | Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010) |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57R65 | Surgery and handlebodies |
Citations:
Zbl 0081.39202; Zbl 0115.17304; Zbl 0345.57001; Zbl 0854.57002; Zbl 0395.57002; Zbl 1072.57005References:
| [1] | DOI: 10.2307/1970041 · Zbl 0081.39202 · doi:10.2307/1970041 |
| [2] | Hempel J., Ann. of Math. Studies, No. 86, in: 3-Manifolds (1976) |
| [3] | Rolfsen D., Mathematics Lecture Series, No. 7, in: Knots and Links (1976) |
| [4] | Myers R., Proc. Amer. Math. Soc. 72 pp 397– |
| [5] | Costich O. L., Proc. Amer. Math. Soc. 28 pp 295– |
| [6] | DOI: 10.2307/1970373 · Zbl 0106.37102 · doi:10.2307/1970373 |
| [7] | DOI: 10.4153/CJM-1960-045-7 · Zbl 0108.36101 · doi:10.4153/CJM-1960-045-7 |
| [8] | DOI: 10.1073/pnas.10.1.6 · doi:10.1073/pnas.10.1.6 |
| [9] | DOI: 10.1007/978-1-4612-9906-6 · doi:10.1007/978-1-4612-9906-6 |
| [10] | Lickorish W. B. R., Michigan Math. J. 36 pp 345– |
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