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

By Sergei Matveev

ISBN-10: 3662051028

ISBN-13: 9783662051023

ISBN-10: 3662051044

ISBN-13: 9783662051047

From the experiences of the first edition:

"This e-book offers a accomplished and certain account of alternative themes in algorithmic three-d topology, culminating with the popularity process for Haken manifolds and together with the up to date leads to machine enumeration of 3-manifolds. Originating from lecture notes of assorted classes given via the writer over a decade, the ebook 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 odd viewpoint of certain polyhedra and exact spines of 3-manifolds. This selection contributes to maintain the extent of the exposition fairly effortless.

In end, the reviewer subscribes to the citation from the again disguise: "the ebook fills a niche within the current literature and may turn into a typical 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 further. particularly, in bankruptcy 7 a number of new sections pertaining to functions of the pc application "3-Manifold Recognizer" were integrated.

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Extra info for Algorithmic Topology and Classification of 3-Manifolds

Sample text

Any two simple spines of the same manifold M are bubble equivalent. J p )II Fig. 23. Removing a 2-cell from the free boundary of a proper ball is a bubble equivalence Proof. Let K c M be a simplicial complex. Assign to K a simple polyhedron W(K) as follows: replace each vertex by a handle of index zero (ball), each edge by a handle of index one (beam) and each triangle by a handle of index two (plate). Then W(K) is defined as the union of the boundaries of all these handles. 13. :,,BW(K); (2) If K \, L, then W(K{,,0B W (L).

It means that after performing a few moves T±l, U±l we would lose the control over attached bubbles, and the bubble move would be essentially equivalent to taking the one-point union with a 2-dimensional sphere. 6. The following lemma tells us that any transient move of a simple polyhedron can be realized by a sequence of moves T, U, L and their inverses. 14. (2-cell shifting) Let P be a simple polyhedron and f, g : 51 ---+ P two homotopic curves in general position. Then the simple polyhedra Q1 = P Uf D2 and Q2 = PUg D2 are (T, U, L)-equivalent.

29 is a composition of moves T±1, U. 30 we may assume that they are situated far from the place where the move is performed. Finally, we get Q with several superfluous loops. 31. ~~. 11':· 47 :. t;plo··~-~ -- Fig. 43. 4 Zeeman's Collapsing Conjecture The following innocuous-looking statement is known as the Zeeman conjecture [133]. Conjecture (ZC). Let X be a contractible polyhedron of dimension::::: 2. Then X x I is collapsible. { *}, i. , X can be collapsed to a point. Every collapsible polyhedron is contractible (it means that the identity map X ---+ X is homotopic to a constant map p: X ---+ Xo C X).

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Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev


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