Download e-book for iPad: Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev

By Sergei Matveev

ISBN-10: 3662051028

ISBN-13: 9783662051023

ISBN-10: 3662051044

ISBN-13: 9783662051047

From the studies of the first edition:

"This ebook offers a finished and precise account of alternative subject matters in algorithmic third-dimensional topology, culminating with the popularity strategy for Haken manifolds and together with the updated ends up in laptop enumeration of 3-manifolds. Originating from lecture notes of varied classes given by way of the writer over a decade, the ebook is meant to mix the pedagogical method of a graduate textbook (without routines) with the completeness and reliability of a learn monograph…

All the cloth, with few exceptions, is gifted from the atypical standpoint of exact polyhedra and specified spines of 3-manifolds. This selection contributes to maintain the extent of the exposition relatively user-friendly.

In end, the reviewer subscribes to the citation from the again hide: "the publication fills a spot within the current literature and should turn into a regular 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 further. particularly, in bankruptcy 7 a number of new sections referring to purposes of the pc application "3-Manifold Recognizer" were integrated.

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Extra resources for Algorithmic Topology and Classification of 3-Manifolds

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Let F be a nonorientable surface with connected boundary such that X(F) = X(H). Present Hand iI respectively as an orient able twisted I-bundle FxI and a trivial I-bundle F x lover F. Then both of them collapse to F with a thin solid tube running along of. See Fig. 37 for the genus one case when H is a solid torus and if is a solid Klein bottle. To get a common simple spine of H, iI, we collapse the tube onto a simple subpolyhedron. I Fig. 37. 20. If K is a simplicial complex, then W(K) is well-defined up to (T, U, L) -equivalence.

37. We say that P2 dominates PI (notation: Pl :::S P2 ) if there exists an embedding i: P l x I ---+ P2 X I such that the following holds: 1. The restriction of i onto aPl x I is jiber-preserving and projects to the identification homeomorphism id. This means that for any (x, t) E aPl x I we have i(:r, t) E {id(:z:)} x I. 2. P2 X I \,i(Pt x 1). 38. 37. For simplicity, we consider I-dimensional special polyhedra, i. , graphs with vertices of valence 1 and 3. Let P l and P2 be two trees such that each of them consists of 5 segments and spans the union aPl = ap2 = Au B U CUD of four points in two different ways.

The proof is the same. Let us assign to a given simplicial complex K2 a simple polyhedron W(K). The assignment is done by means of a construction. Given a simplicial complex K2, let IK(I) I denote its I-skeleton. The construction is carried out in four steps. 3 Special Polyhedra Which are not Spines 39 Fig. 35. Realization of the "triangle" Reidemeister move 1. Choose an orient able or nonorientable handlebody H3 such that X(H3) = X(IK(1)I). Note that IK(l)1 and H are homotopy equivalent. 2. Choose a homotopy equivalence 'P: IK(1)1 ---+ H.

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