Prime Decomposition of Three-Dimensional Manifolds into Boundary Connected Sum

In 2003 Matveev suggested a new version of the Diamond Lemma suitable for topological applications. We apply this result to different situations and get a new conceptual proof of theorem on decomposition of three-dimensional manifolds into boundary connected sum of prime components. 1. Introduction Since 2003 Matveev [1–3] had suggested a new version of the Diamond Lemma [4] of great importance for various fields of mathematics, which is suitable for and efficient solving topological problems. In this paper we apply this result to get a new conceptual proof of theorem on decomposition of three-dimensional manifolds into boundary connected sum of prime components. 2. Definition, Lemma, and -Irreducible Manifolds Diamond Lemma (see [5]). If an oriented graph has the properties (FP) and (MF), then each of its vertices has a unique root. Definition 1. Let be a proper disk in a compact connected 3-manifold . A disk reduction of along consists in cutting along the disk . Its result is a new manifold . We apply nontrivial disk reductions to a given manifold as many times as possible. If this process stops, then we obtain a set of new manifolds, which is called a root of .(1)If the disk is splitting, the manifold is obtained by gluing the manifolds and together along disks on their boundaries. Then is called a boundary connected sum of the manifolds and and denoted by .(2)If the disk is nonsplitting, then is also connected. Definition 2. is said to be irreducible if every properly embedded disk in is trivial. Lemma 3 (see [6]). Let be a -irreducible manifold. Let be the manifold obtained from by attaching a 1-handle to make the boundary connected. Then is a prime which is not irreducible. 3. Proof of Theorem？？4 Theorem 4. Any connected irreducible compact 3-manifold different from a ball and with nonempty boundary is homeomorphism to a boundary connected sum of prime manifolds. All the summands are defined uniquely up to reordering and, if is non-Orientale, replacing solid tori by solid Klein bottles . We apply the universal scheme [4] in two stages. First, by considering reductions along all disks we establish uniqueness of the -irreducible manifolds . Then we restrict ourselves to reductions only along splitting disks and by lemma [3] complete the proof of the theorem. We construct the graph [5] whose vertices are compact connected irreducible manifolds, considered up to addition or deletion of three-dimensional balls. The edges of the graph correspond to reductions along both splitting and nonsplitting disks. Lemma 5 (see [7]). Each essential disks reduction