Roots in 3-manifold topology C HOG-ANGELONI S MATVEEV Let C be some class of objects equipped with a set of simplifying moves. When we apply these to a given object M ∈ C as long as possible, we get a root of M .Our main result is that under certain conditions the root of any object exists and is unique. We apply this result to different situations and get several new results and new proofs of known results. Among them there are a new proof of the Kneser-Milnor prime decomposition theorem for 3-manifolds and different versions of this theorem for cobordisms, knotted graphs, and orbifolds. 57N10; 57M992 Definition, existence and uniqueness of a rootLet Γ be an oriented graph and e an edge of Γ with initial vertex v and terminal vertex w. We will call the transition from v to w an edge move on v.Definition 1 A vertex R(v) of Γ is a root of v, if the following holds:(1) R(v) can be obtained from v by edge moves.(2) R(v) admits no further edge moves.Recall that a set A is well ordered if any subset of A has a least element. Basic examples are the set of non-negative integers N 0 and its power N k 0 with lexicographical order.Definition 2 Let Γ be an oriented graph with vertex set V(Γ) and A a well ordered set. Then a map c : V(Γ) → A is called a complexity function, if for any edge e of Γ with vertices v, w and orientation from v to w we have c(v) > c(w).Definition 3 Let Γ be an oriented graph. Then two edges e and d of Γ with the same initial vertex v are called elementary equivalent, if their endpoints have a common root. They are called equivalent (notation: e ∼ d ), if there is a sequence of edges e = e 1 , e 2 , . . . , e n = d such that the edges e i and e i+1 are elementary equivalent for all i, 1 ≤ i < n.Definition 4 Let Γ be an oriented graph. We say that Γ possesses property (CF) if it admits a complexity function. Γ possesses property (EE) if any two edges of Γ with common initial vertex are equivalent.It turns out that property (CF) guarantees existence, while property (EE) guarantees uniqueness of the root.Theorem 1 Let Γ be an oriented graph possessing properties (CF) and (EE). Then any vertex has a unique root.Proof Existence Let v be a vertex of Γ. Denote by X the set of all vertices of Γ which can be obtained from v by edge moves. By property (CF), there is a complexity function c : V(Γ) → A. Since A is well ordered, the set c(X) has a least element a 0 . Then any vertex in c −1 (a 0 ) is a root of v.Uniqueness Assume that v is a least counterexample, ie, v has two different roots u = w and c(v) ≤ c(v ) for any vertex v having more than one root. Let e respectively d be the first edge of an oriented edge path from v toward u respectively w. By property (EE), we have a sequence e = e 1 , e 2 , . . . , e n = d such that the edges e i and e i+1 are elementary equivalent for all i, 1 ≤ i < n. Hence, their endpoints v i , v i+1 have a common root r i . As c(v i ) < c(v) for all i, that root is in fact unique. Thus u = r 1 = · · · = r n = w which is a contradiction.The following sections are devoted to app...
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