By Sergei Matveev
From the studies of the first edition:
"This e-book offers a finished and certain account of alternative issues in algorithmic three-dimensional topology, culminating with the popularity technique for Haken manifolds and together with the up to date ends up in laptop enumeration of 3-manifolds. Originating from lecture notes of assorted classes given via the writer over a decade, the publication is meant to mix the pedagogical process of a graduate textbook (without routines) with the completeness and reliability of a examine monograph…
All the cloth, with few exceptions, is gifted from the abnormal viewpoint of distinct polyhedra and precise spines of 3-manifolds. This selection contributes to maintain the extent of the exposition quite effortless.
In end, the reviewer subscribes to the citation from the again conceal: "the booklet fills a niche within the present literature and should develop into a customary reference for algorithmic three-dimensional topology either for graduate scholars and researchers".
Zentralblatt für Mathematik 2004
For this 2nd version, new effects, new proofs, and commentaries for a greater orientation of the reader were additional. specifically, in bankruptcy 7 numerous new sections pertaining to purposes of the pc application "3-Manifold Recognizer" were incorporated.
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Additional resources for Algorithmic Topology and Classification of 3-Manifolds
16. The only difference is that there is no 3-manifold, where D 1 , D 2 , and the trace of the homotopy between the curves could bound a proper 3-ball. Let ft : 51 ---+ P be a homotopy between f and g. Define a map F : 51 x I ---+ P x I by the rule F(x, t) = (ft(x), t). We say that ft is regular at a point (x, T) E 51 X I, if the restriction of F to a neighborhood of this point is an embedding. We call a moment of time T singular, if F is not regular at some point (x, T) or the curve iT(5 1) is not in general position.
These extra arches can be removed by moves T ±l and L±l, see Fig. 29. 11. 8 and by the assumption on the number of true vertices of P1 , P2 , every V-move is a composition of T-moves. I T- 1 T 1 Fig. 31. 5. 3 Special Polyhedra Which are not Spines We have seen in Sect. 18) that there exist special polyhedra which are not spines of any 3-manifold. Let us describe a systematic way to construct such examples. We start with any special polyhedron P and transform it to produce a new special polyhedron P l that does not embed into a 3-manifold.
2 Elementary Moves on Special Spines a b c •• •• •• • •• Fig. 21. Events in neighborhoods of singular moments Fig. 22. The bubble move 23 24 1 Simple and Special Polyhedra among them one disc D and add it to P. We get a new simple subpolyhedron P'cM. 17. The transition from P to P' is called a bubble move at x and denoted by B. Let us describe a few properties of the bubble move. First, it is an ambient move; one can apply it only to subpolyhedra of 3-manifolds. Of course, one can define a bubble move for abstract simple polyhedra, but we do not need that.
Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev