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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An algebraic characterization of connected sum factors of closed $ 3$-manifolds


Author: W. H. Row
Journal: Trans. Amer. Math. Soc. 250 (1979), 347-356
MSC: Primary 57M25; Secondary 57N10
MathSciNet review: 530060
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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. $ K\,(M)\, = \,K\,(N)$ iff M is homeomorphic to N.

Given M, there exists a knot group $ {G_M}$ of M that serves to characterize M in the following sense.

Corollary 2. $ {G_M}$ is realized in N and $ {G_N}$, 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: http://dx.doi.org/10.1090/S0002-9947-1979-0530060-2
PII: S 0002-9947(1979)0530060-2
Keywords: Connected sum, knot group, submanifold group, cube-with-a-knotted-hole, $ {P^2}$-irreducible
Article copyright: © Copyright 1979 American Mathematical Society