Download Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev PDF

By Sergei Matveev

From the studies of the first edition:

"This publication offers a finished and particular account of alternative issues in algorithmic three-dimensional topology, culminating with the popularity process for Haken manifolds and together with the updated ends up in laptop enumeration of 3-manifolds. Originating from lecture notes of assorted classes given by means of the writer over a decade, the booklet is meant to mix the pedagogical procedure 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 unusual viewpoint of exact polyhedra and certain spines of 3-manifolds. This selection contributes to maintain the extent of the exposition rather straight forward.

In end, the reviewer subscribes to the citation from the again conceal: "the ebook fills a spot within the present literature and should turn into a customary reference for algorithmic third-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 extra. specifically, in bankruptcy 7 a number of new sections referring to functions of the pc application "3-Manifold Recognizer" were incorporated.

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Extra info for Algorithmic Topology and Classification of 3-Manifolds

Example text

Creating a loop by the moves U, U, T- 1 A5 coincides with the U-move. For a realization of A3 we assume that two of the three arcs of It that take part in the modification are fixed, and that the corresponding parts of D2 are already attached. We use them to express A3 as a composition of T, U- 1 , U, T- 1 , see Fig. 35. 16. 14 remains true if in its statement one replaces the disc D2 by an arbitrary simple polyhedron whose boundary consists of a 0 circle. The proof is the same. Let us assign to a given simplicial complex K2 a simple polyhedron W(K).

28. First steps create an arch with a disc inside it, the last one converts the disc to a bubble. 26. Suppose two marked polyhedra (Q1, md, (Q2, m2) in a 3-manifold Mare (T,L,m)-equivalent. - A(Q2,m2). Proof. Arguing by induction, we can suppose that (Q2 , m2) is obtained from (Q1 , m1) by a single move. The case of moves T±1 , L±l performed far from the mark is obvious: one needs only perform the moves far from the arch. 2 Elementary Moves on Special Spines 29 Fig. 28. 25 twice: first remove the arch and the bubble, then re-create the bubble with the arch in the other position.

3) Suppose that H3 is orientable. 18 that any two homotopy equivalences 'P, 'P': IK(1) I -+ H3 differ by a homeomorphism H3 -+ H3. This means that thc corresponding blow-ups are equivalent. 19) we get equivalent blow-ups. By (1) and (2), the (T, U, L)-type of W(K) does not depend on the choice of a spine. It follows that W(K) does not depend on the choice of the handlebody. We have shown that any two blow-ups of K are (T, U, L)-equivalent. 21. L. Then any sequence S1, 82, ... , 8 n of elementary simplicial collapses and expansions of dimension::; 3 transforming K into L can be rearranged so that it consists of simplicial transient moves and deformations of dimension::; 2.

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