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.

**[B]**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**[B-M]**R. H. Bing and J. M. Martin,*Cubes with knotted holes*, Trans. Amer. Math. Soc.**155**(1971), 217–231. MR**0278287**, https://doi.org/10.1090/S0002-9947-1971-0278287-4**[C]**A. C. Conner,*An algebraic characterization of*3-*manifolds*, Notices Amer. Math. Soc.**17**(1970), 266 Abstract #672-635.**[E]**D. B. A. Epstein,*Projective planes in 3-manifolds*, Proc. London Math. Soc. (3)**11**(1961), 469–484. MR**0152997**, https://doi.org/10.1112/plms/s3-11.1.469**[F]**C. D. Feustel,*A generalization of Kneser’s conjecture*, Pacific J. Math.**46**(1973), 123–130. MR**0328908****[Ha]**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****[Hei]**Wolfgang Heil,*On 𝑃²-irreducible 3-manifolds*, Bull. Amer. Math. Soc.**75**(1969), 772–775. MR**0251731**, https://doi.org/10.1090/S0002-9904-1969-12283-4**[Hem]**John Hempel,*3-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619****[J-M]**William Jaco and Robert Myers,*An algebraic characterization of closed*3-*manifolds*, Notices Amer. Math. Soc.**24**(1977), A-263 Abstract #77T-G39.**[W1]**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**[W2]**Friedhelm Waldhausen,*Eine Verallgemeinerung des Schleifensatzes*, Topology**6**(1967), 501–504 (German). MR**0220300**, https://doi.org/10.1016/0040-9383(67)90007-9

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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