An algebraic characterization of connected sum factors of closed -manifolds

Author:
W. H. Row

Journal:
Trans. Amer. Math. Soc. **250** (1979), 347-356

MSC:
Primary 57M25; Secondary 57N10

DOI:
https://doi.org/10.1090/S0002-9947-1979-0530060-2

MathSciNet review:
530060

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *M* and *N* be closed connected 3-manifolds. A *knot group* of *M* is the fundamental group of the complement of a tame simple closed curve in *M*. Denote the set of knot groups of *M* by *K*(*M*). A knot group *G* of *M* is *realized* in *N* if *G* is the fundamental group of a compact submanifold of *N* with connected boundary.

Theorem. *Every knot group of N is realized in M iff N is a connected sum factor of M*.

Corollary 1. iff *M* is homeomorphic to *N*.

Given *M*, there exists a knot group of *M* that serves to characterize *M* in the following sense.

Corollary 2. *is realized in N and* , *is realized in M iff M is homeomorphic to N*.

Our proof depends heavily on the work of Bing, Feustal, Haken, and Waldhausen in the 1960s and early 1970s. A. C. Conner announced Corollary 1 for orientable 3-manifolds in 1969 which Jaco and Myers have recently obtained using different techniques.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0530060-2

Keywords:
Connected sum,
knot group,
submanifold group,
cube-with-a-knotted-hole,
-irreducible

Article copyright:
© Copyright 1979
American Mathematical Society