Simple knots in compact, orientable -manifolds

Author:
Robert Myers

Journal:
Trans. Amer. Math. Soc. **273** (1982), 75-91

MSC:
Primary 57N10; Secondary 57M25, 57M40

DOI:
https://doi.org/10.1090/S0002-9947-1982-0664030-0

MathSciNet review:
664030

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 boundary-irreducible and contains no non-boundary-parallel, 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.

**[1]**R. H. Bing,*Necessary and sufficient conditions that a 3-manifold be 𝑆³*, Ann. of Math. (2)**68**(1958), 17–37. MR**0095471**, https://doi.org/10.2307/1970041**[2]**R. H. Fox,*Recent development of knot theory at Princeton*, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R. I., 1952, pp. 453–457. MR**0048023****[3]**Wolfgang Haken,*Some results on surfaces in 3-manifolds*, Studies in Modern Topology, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1968, pp. 39–98. MR**0224071****[4]**John Hempel,*3-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619****[5]**William Jaco,*Three-manifolds with fundamental group a free product*, Bull. Amer. Math. Soc.**75**(1969), 972–977. MR**0243531**, https://doi.org/10.1090/S0002-9904-1969-12320-7**[6]**William Jaco,*Lectures on three-manifold topology*, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR**565450****[7]**William Jaco and Robert Myers,*An algebraic determination of closed orientable 3-manifolds*, Trans. Amer. Math. Soc.**253**(1979), 149–170. MR**536940**, https://doi.org/10.1090/S0002-9947-1979-0536940-6**[8]**William H. Jaco and Peter B. Shalen,*Seifert fibered spaces in 3-manifolds*, Mem. Amer. Math. Soc.**21**(1979), no. 220, viii+192. MR**539411**, https://doi.org/10.1090/memo/0220**[9]**Klaus Johannson,*Homotopy equivalences of 3-manifolds with boundaries*, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR**551744****[10]**D. R. McMillan Jr.,*On homologically trivial 3-manifolds*, Trans. Amer. Math. Soc.**98**(1961), 350–367. MR**0120639**, https://doi.org/10.1090/S0002-9947-1961-0120639-0**[11]**J. Milnor,*A unique decomposition theorem for 3-manifolds*, Amer. J. Math.**84**(1962), 1–7. MR**0142125**, https://doi.org/10.2307/2372800**[12]**José M. Montesinos,*Surgery on links and double branched covers of 𝑆³*, Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Princeton Univ. Press, Princeton, N.J., 1975, pp. 227–259. Ann. of Math. Studies, No. 84. MR**0380802****[13]**R. Myers,*Homology cobordisms, link concordances, and hyperbolic**-manifolds*(preprint).**[14]**W. H. Row,*An algebraic characterization of connected sum factors of closed 3-manifolds*, Trans. Amer. Math. Soc.**250**(1979), 347–356. MR**530060**, https://doi.org/10.1090/S0002-9947-1979-0530060-2**[15]**Horst Schubert,*Knoten und Vollringe*, Acta Math.**90**(1953), 131–286 (German). MR**0072482**, https://doi.org/10.1007/BF02392437**[16]**H. Seifert,*Schlingknoten*, Math. Z.**52**(1949), 62–80 (German). MR**0031732**, https://doi.org/10.1007/BF02230685**[17]**W. Thurston,*The geometry and topology of**-manifolds*, lecture notes, Princeton.**[18]**Friedhelm Waldhausen,*Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II*, Invent. Math. 3 (1967), 308–333; ibid.**4**(1967), 87–117 (German). MR**0235576**, https://doi.org/10.1007/BF01402956**[19]**Friedhelm Waldhausen,*On irreducible 3-manifolds which are sufficiently large*, Ann. of Math. (2)**87**(1968), 56–88. MR**0224099**, https://doi.org/10.2307/1970594**[20]**Wilbur Whitten,*Groups and manifolds characterizing links*, Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Princeton Univ. Press, Princeton, N.J., 1975, pp. 63–84. Ann. of Math. Studies, No. 84. MR**0385840**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57N10,
57M25,
57M40

Retrieve articles in all journals with MSC: 57N10, 57M25, 57M40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0664030-0

Keywords:
-manifold,
knot,
simple knot,
simple -manifold,
semisimple -manifold,
hyperbolic -manifold,
knot group,
Poincaré Conjecture

Article copyright:
© Copyright 1982
American Mathematical Society