Algorithmic topology and classification of 3-manifolds by Sergei Matveev

By Sergei Matveev

From the stories of the first edition:

"This e-book offers a finished and certain account of other subject matters in algorithmic three-d topology, culminating with the popularity strategy for Haken manifolds and together with the up to date leads to computing device enumeration of 3-manifolds. Originating from lecture notes of assorted classes given via the writer over a decade, the booklet is meant to mix the pedagogical strategy of a graduate textbook (without routines) with the completeness and reliability of a learn monograph…

All the fabric, with few exceptions, is gifted from the ordinary perspective of targeted polyhedra and specific spines of 3-manifolds. This selection contributes to maintain the extent of the exposition particularly hassle-free.

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

Zentralblatt f?r Mathematik 2004

For this 2nd version, new effects, new proofs, and commentaries for a greater orientation of the reader were extra. particularly, in bankruptcy 7 a number of new sections referring to purposes of the pc application "3-Manifold Recognizer" were incorporated.

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Sample text

The problem is whether it is always possible to continue the collapse so that we do not become stuck before we reach a point. 32. A polyhedron X is called 1-collapsible, if X × I {∗}. In these terms Zeeman’s Conjecture can be reformulated as follows: Any contractible 2-dimensional polyhedron is 1-collapsible. One can easily show that the Bing House and Abalone are 1-collapsible. 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].

We get a new special spine P1 of the same manifold. 28. The transition from P to P1 will be called creation of an arch with a membrane. 29. An arch with a membrane can be created by means of one move L. Proof. See Fig. 29. 30. Let P1 and P2 be special spines of the same manifold M with at least two true vertices each. Then they are T -equivalent. Proof. 27, one can pass from P1 to P2 by moves T ±1 , L, which preserve the property of a spine of having only disc 2-components, and obius 2-components.

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 , m1 ), (Q2 , m2 ) in a 3-maT,L nifold M are (T, L, m)-equivalent. Then A(Q1 , m1 ) ∼ 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±1 performed far from the mark is obvious: one needs only perform the moves far from the arch. 25 twice: First remove the arch and the bubble, then re-create the bubble with the arch in the other position.

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