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

Row, W. H.

doi:10.1090/S0002-9947-1979-0530060-2

An algebraic characterization of connected sum factors of closed $3$-manifolds
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by W. H. Row PDF

Trans. Amer. Math. Soc. 250 (1979), 347-356 Request permission

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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