Sergei Matveev's Algorithmic Topology and Classification of 3-Manifolds PDF

By Sergei Matveev

ISBN-10: 3662051028

ISBN-13: 9783662051023

ISBN-10: 3662051044

ISBN-13: 9783662051047

From the experiences of the first edition:

"This booklet offers a entire and exact account of alternative issues in algorithmic three-dimensional topology, culminating with the popularity approach for Haken manifolds and together with the updated leads to desktop 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 method of a graduate textbook (without workouts) with the completeness and reliability of a examine monograph…

All the fabric, with few exceptions, is gifted from the extraordinary standpoint of specific polyhedra and precise spines of 3-manifolds. This selection contributes to maintain the extent of the exposition particularly uncomplicated.

In end, the reviewer subscribes to the citation from the again conceal: "the booklet fills a spot within the present literature and may develop into a regular reference for algorithmic three-d topology either for graduate scholars and researchers".

Zentralblatt für Mathematik 2004

For this 2nd variation, new effects, new proofs, and commentaries for a greater orientation of the reader were further. specifically, in bankruptcy 7 numerous new sections relating functions of the pc software "3-Manifold Recognizer" were integrated.

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Taking a contractible polyhedron X and assuming ZC, we have X~X x I~{ *}. 35. 34 remains true if we restrict ZC to the class of special polyhedra. Only minor modifications in the proof are needed. For proving ZC :::} PC one should take not an arbitrary but a special spine X of M, and for proving ZC :::} AC one should first 3-deform X to a special polyhedron and only then take the product with I. We may conclude that (1) ZC for special spines implies PC; (2) ZC for unthickenable special polyhedra implies AC.

The elementary move U on a simple polyhedron P consists in removing a proper butterfly E C P and replacing it by E u , see Fig. 33. u :=-=:U- i I Fig. 33. The U-move Notice that U increases the number of true vertices in a polyhedron by one, and that the Eu does not embed into R3. 9. Let P, Q be special polyhedra. Then P~Q if and only if one can transform Pinto Q by a finite sequence of moves T±I, U±1 . 11 below. 10. Let 2-dimensional polyhedra Xl, X 2 be obtained from a polyhedron Y by attaching discs Dr, D§ with homotopic attaching maps II, h : 51 ---+ Y.

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. Proof. Let Sk be the first 3-dimensional collapse in the sequence. The tetrahedron (j that disappears after 8k (together with its free triangle face 8) has been created by an expansion 8 m , Tn < k.

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Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev

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