Simple knots in compact, orientable -manifolds

Author:
Robert Myers

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

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 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.

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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