A Unique Decomposition Theorem for 3-Manifolds with Connected Boundary

Handlebody Splittings of Compact 3-Manifolds with Boundary

“…A bordered 3-manifold H is said to be a handlebody of genus g iff H is the disk-sum (i.e., the boundary connected-sum) of g copies of the solid-torus D 2 × S 1 (see Gross [3], Swarup [16], etc.). A handlebody of genus g is characterized as a regular neighborhood N (P ; R 3 ) of a connected 1-polyhedron P with Euler characteristic χ(P ) = 1−g in the 3-dimensional Euclidean space R 3 and as an irreducible bordered 3-manifold M with connected boundary whose fundamental group π 1 (M ) is a free group of rank g (see Ochiai [10]).…”

The decomposition of 3-manifolds with several boundary components.

“…One says that a 3-manifold F with connected nonvacuous boundary is A-prime if F is not a 3-cell and whenever F is homeomorphic to a disk sum MAM', either M or AF is a 3-cell. The following theorem is proved by the author in [1]: Theorem 1. Let M be a 3-manifold with connected, nonvacuous boundary.…”

“…Lemma 3 is an extension of Lemma 6 of [1]. One obtains a proof of Lemma 3 by generalizing the steps in the proof of Lemma 6 of [1].…”

“…The proof of uniqueness to be given here will be obtained by generalizing the techniques of [1]. The notation used here reflects the notation used in [1].…”

“…The notation used here reflects the notation used in [1]. One considers here a fixed collection of mutually disjoint m-prime 3-manifold Pi,..., Pn and a fixed iteration of n-1 multi-disk sum operations on them which results in a 3-manifold homeomorphic to M. The next paragraph is concerned with the construction of a 3-manifold M * which is homeomorphic to M and in which it will be convenient to perform cutting and pasting operations.…”