An algebraic characterization of connected sum factors of closed manifolds
Author:
W. H. Row
Journal:
Trans. Amer. Math. Soc. 250 (1979), 347356
MSC:
Primary 57M25; Secondary 57N10
MathSciNet review:
530060
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Abstract: Let M and N be closed connected 3manifolds. 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 3manifolds in 1969 which Jaco and Myers have recently obtained using different techniques.
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 R. H. Bing and J. M. Martin, Cubes with knotted holes, Trans. Amer. Math. Soc. 155 (1971), 217231. MR 0278287 (43:4018a)
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 A. C. Conner, An algebraic characterization of 3manifolds, Notices Amer. Math. Soc. 17 (1970), 266 Abstract #672635.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905300602
PII:
S 00029947(1979)05300602
Keywords:
Connected sum,
knot group,
submanifold group,
cubewithaknottedhole,
irreducible
Article copyright:
© Copyright 1979
American Mathematical Society
