Differential Geometry

Read e-book online Algorithmic topology and classification of 3-manifolds PDF

By Sergei Matveev

ISBN-10: 3540458980

ISBN-13: 9783540458982

From the stories of the first edition:

"This ebook presents a entire and specific account of other subject matters in algorithmic three-d topology, culminating with the popularity technique for Haken manifolds and together with the up to date ends up in desktop enumeration of 3-manifolds. Originating from lecture notes of varied classes given via the writer over a decade, the ebook is meant to mix the pedagogical process of a graduate textbook (without workouts) with the completeness and reliability of a study monograph…

All the fabric, with few exceptions, is gifted from the unusual standpoint of distinct polyhedra and unique spines of 3-manifolds. This selection contributes to maintain the extent of the exposition relatively common.

In end, the reviewer subscribes to the citation from the again hide: "the ebook fills a spot within the present literature and should develop 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 numerous new sections touching on purposes of the pc application "3-Manifold Recognizer" were integrated.

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Extra info for Algorithmic topology and classification of 3-manifolds

Example text

It turns out that only the choice of the neutral disc is important. For instance, moving A and fixed C (see Fig. 32) and moving C and fixed A produce the same result. Therefore, in a neighborhood of a triple point the move α can be made in three different ways. 27. Just to fix terminology, let us say that a special polyhedron P contains a loop if the singular graph SP of P contains a loop such that is the boundary curve of a 2-component of P and has a nontrivial normal bundle. It is evident that P contains a loop if and only if it can be obtained from another special polyhedron by the loop move.

A simple polyhedron P ⊂ M with a mark m is called a marked polyhedron and denoted (P, m). 22. Two marked polyhedra P1 , P2 are (T, L, m)-equivalent T,L,m (notation: P1 ∼ P2 ) if one can pass from one to the other by the following moves: (1) T ±1 - and L±1 -moves carried out far from the mark. This means that the mark must lie outside the fragments of P that are replaced during the moves. (2) m-move consisting of transferring the mark from one 2-cell of a bubble to another 2-cell of the same bubble.

1–3, 11, 12 of [47] for a complete account of the subject. Let X, Y be 2-dimensional polyhedra. (I) Isomorphism of fundamental groups. X ∼ Y if π1 (X) = π1 (Y ). This relation is fairy rough. For example, taking the one-point union with S 2 preserves the fundamental group but increases the Euler characteristic by 1. 22) is essentially an addition of S 2 . (II) The same fundamental group and Euler characteristic. X ∼ Y if π1 (X) = π1 (Y ) and χ(X) = χ(Y ). (III) Homotopy equivalence. Recall that two topological spaces X and Y are homotopy equivalent if there exist maps f : X → Y and g: Y → X such 32 1 Simple and Special Polyhedra that the maps f g : X → X and gf : Y → Y are homotopic to the identity.

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Algorithmic topology and classification of 3-manifolds by Sergei Matveev

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