Simple knots in compact, orientable manifolds
Author:
Robert Myers
Journal:
Trans. Amer. Math. Soc. 273 (1982), 7591
MSC:
Primary 57N10; Secondary 57M25, 57M40
MathSciNet review:
664030
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Abstract: A simple closed curve in the interior of a compact, orientable manifold is called a simple knot if the closure of the complement of a regular neighborhood of in is irreducible and boundaryirreducible and contains no nonboundaryparallel, properly embedded, incompressible annuli or tori. In this paper it is shown that every compact, orientable manifold such that contains no spheres contains a simple knot (and thus, from work of Thurston, a knot whose complement is hyperbolic). This result is used to prove that such a manifold is completely determined by its set of knot groups, i.e, the set of groups as ranges over all the simple closed curves in . In addition, it is proven that a closed manifold is homeomorphic to if and only if every simple closed curve in shrinks to a point inside a connected sum of graph manifolds and cells.
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 R. H. Bing, Necessary and sufficient conditions that a manifold be , Ann. of Math. 68 (1958), 1737. MR 0095471 (20:1973)
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 [4]
 J. Hempel, manifolds, Ann. of Math. Studies, no. 86, Princeton Univ. Press, Princeton, N. J., 1976. MR 0415619 (54:3702)
 [5]
 W. Jaco, Threemanifolds with fundamental group a free product, Bull. Amer. Math. Soc. 75 (1969), 972977. MR 0243531 (39:4852)
 [6]
 , Lectures on threemanifold topology, CBMS Regional Conference Series in Math. No. 43, Amer. Math. Soc., Providence, R. I., 1980. MR 565450 (81k:57009)
 [7]
 W. Jaco and R. Myers, An algebraic determination of closed, orientable manifolds, Trans. Amer. Math. Soc. 253 (1979), 149170. MR 536940 (80g:57012)
 [8]
 W. Jaco and P. Shalen, Seifert fibered spaces in manifolds, Mem. Amer. Math. Soc. No. 220 (1979). MR 539411 (81c:57010)
 [9]
 K. Johannson, Homotopy equivalences of manifolds with boundaries, Lecture Notes in Math., vol. 761, SpringerVerlag, Berlin and New York, 1979. MR 551744 (82c:57005)
 [10]
 D. R. McMillan, On homologically trivial manifolds, Trans. Amer. Math. Soc., 98 (1961), 350367. MR 0120639 (22:11389)
 [11]
 J. Milnor, A unique factorization theorem for manifolds, Amer. J. Math. 84 (1962), 17. MR 0142125 (25:5518)
 [12]
 J. M. Montesinos, Surgery on links and double branched covers of , Knots, Groups, and Manifolds, Ann. of Math. Studies, no. 84, Princeton Univ. Press, Princeton, N. J., 1975, pp. 227260. MR 0380802 (52:1699)
 [13]
 R. Myers, Homology cobordisms, link concordances, and hyperbolic manifolds (preprint).
 [14]
 W. H. Row, An algebraic characterization of connected sum factors of closed manifolds, Trans. Amer. Math. Soc. 250 (1979), 347356. MR 530060 (80g:57009)
 [15]
 H. Schubert, Knoten und Vollringe, Acta. Math. 90 (1953), 131286. MR 0072482 (17:291d)
 [16]
 H. Seifert, Schlingknoten, Math. Z. 52 (1949), 6280. MR 0031732 (11:196e)
 [17]
 W. Thurston, The geometry and topology of manifolds, lecture notes, Princeton.
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 F. Waldhausen, Eine Klasse von dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308333; ibid 4 (1967), 87117. MR 0235576 (38:3880)
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 , On irreducible manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 5688. MR 0224099 (36:7146)
 [20]
 W. Whitten, Groups and manifolds characterizing links, Knots, Groups, and Manifolds, Ann. of Math. Studies, no. 84, Princeton Univ. Press, Princeton, N. J., 1975, pp. 6386. MR 0385840 (52:6699)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206640300
PII:
S 00029947(1982)06640300
Keywords:
manifold,
knot,
simple knot,
simple manifold,
semisimple manifold,
hyperbolic manifold,
knot group,
Poincaré Conjecture
Article copyright:
© Copyright 1982
American Mathematical Society
