Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev

By Sergei Matveev

From the stories of the first version: "This ebook presents a accomplished and distinct account of other themes in algorithmic three-d topology, culminating with the popularity process for Haken manifolds and together with the up to date leads to laptop enumeration of 3-manifolds. Originating from lecture notes of varied classes given by way of the writer over a decade, the publication is meant to mix the pedagogical procedure of a graduate textbook (without routines) with the completeness and reliability of a learn monograph… all of the fabric, with few exceptions, is gifted from the bizarre viewpoint of distinct polyhedra and detailed spines of 3-manifolds. This selection contributes to maintain the extent of the exposition quite ordinary. In end, the reviewer subscribes to the citation from the again conceal: "the booklet fills a spot within the current literature and should turn into a typical reference for algorithmic third-dimensional topology either for graduate scholars and researchers". Zentralblatt f?r Mathematik 2004 For this second variation, new effects, new proofs, and commentaries for a greater orientation of the reader were further. particularly, in bankruptcy 7 a number of new sections pertaining to purposes of the pc application "3-Manifold Recognizer" were integrated.

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If x lies on a triple line of P , then S(x) is homeomorphic to Y × I, where Y is a wedge of three segments with a common endpoint. 2 Elementary Moves on Special Spines 23 Fig. 22. The bubble move 3. If x is a true vertex, then S(x) is a butterfly. Evidently, S(x) is a simple polyhedron whose boundary ∂S(x) decomposes the sphere ∂N (x) into 2, 3, or 4 discs, depending on the type of x. Choose from them one disc D and add it to P . We get a new simple subpolyhedron P ⊂ M. 17. The transition from P to P is called a bubble move at x and denoted by B.

Let Q be obtained from P by attaching a Bing membrane T,L B 2 . Then Q ∼ P . Proof. First we shift B 2 into a neighborhood of a nonsingular point of P . 15. Then we move the disc bounded by ∂B 2 into the upper tube to get an arch with the membrane, and destroy the arch. 29 shows how it can be done by the move L−1 . Similarly, we destroy the second arch, see Fig. 38. 22. Blow-ups of 2-dimensional complexes possess the following properties: (1) If the underlying polyhedron |K| of a complex K is simple, then any blowup W (K) of |K| is (T, L)-equivalent to |K|.

For a recent detailed account of the contemporary status of Zeeman’s Conjecture, Andrews–Curtis Conjecture and other conjectures in low-dimensional topology and combinatorial group theory we refer the reader to the fundamental book [47]. Let us recall now the famous Poincar´e Conjecture. 3 Special Polyhedra Which are not Spines 47 Conjecture (PC). Any simply connected closed 3-manifold is homeomorphic to S 3 . Since a closed 3-manifold is simply connected if and only if it is homotopy equivalent to S 3 , one can reformulate PC as follows: Any homotopy 3-sphere is S 3 .

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