Algorithmic Topology and Classification of 3-Manifolds

1. Front Matter

   Pages I-XII

   PDF

2. Simple and Special Polyhedra

   • Sergei Matveev

   Pages 1-58

3. Complexity Theory of 3-Manifolds

   • Sergei Matveev

   Pages 59-106

4. Haken Theory of Normal Surfaces

   • Sergei Matveev

   Pages 107-145

5. Applications of the Theory of Normal Surfaces

   • Sergei Matveev

   Pages 147-191

6. Algorithmic Recognition of S ³

   • Sergei Matveev

   Pages 193-214

7. Classification of Haken 3-Manifolds

   • Sergei Matveev

   Pages 215-328

8. Computer Implementation

   • Sergei Matveev

   Pages 329-366

9. The Turaev-Viro Invariants

   • Sergei Matveev

   Pages 367-403

10. Appendix

    • Sergei Matveev

    Pages 405-466

11. Back Matter

    Pages 467-478

    PDF

The book is devoted to algorithmic low-dimensional topology. This branch of mathematics has recently been undergoing an intense development. On the one hand, the exponential advancement of computer technologies has made it possible to conduct sophisticated computer experiments and to implement algorithmic solutions, which have in turn provided a motivation to search for new and better algorithms. On the other hand, low-dimensional topology has received an additional boost because of the discovery of numerous connections with theoretical physics. There is also another deep reason why algorithmic topology has received a lot of attention. It is that a search for algorithmic solutions generally proves to be a rich source of well-stated mathematical problems. Speaking out of my experience, it seems that an orientation towards "how to" rather than just "how is" serves as a probing stone for choosing among possible directions of research - much like problems in mechanics led once to the development of calculus.

• 3-manifold
• Homeomorphism
• algorithmic recognition
• algorithms
• ants
• classification
• complexity
• special spine
• sufficiently large manifold

From the reviews:

"This book provides a comprehensive and detailed account of different topics in algorithmic 3-dimensiona] topology, culminating with the recognition procedure for Haken manifolds and including the up-to-date results in Computer enumeration of 3-manifolds. Originating from lecture notes of various courses given by the author over a decade, the book is intended to combine the pedagogical approach of a graduate textbook (without exercises) with the completeness and reliability of a research monograph. This motivates the about 400 pages length of the main text.

All the material, with few exceptions, is presented froni the peculiar point of view of special polyhedra and special spines of 3-manifolds. This choice contributes to keep the level of the exposilion really elementary. ….

In conclusion, the reviewer subscribes to the quotation from the back cover: "the book fills a gap in the existing literature and will become a standard reference for algorithmic 3-dimensional topology both for graduate students and researchers".

Riccardo Piergallini (Camerino), Zentralblatt für Mathematik 1048 (2004)

"The purpose of the book is to present a detailed overview of the algorithmis aspects of 3-manifold topology. As noted by the author, the book is largely self-contained, though basic topology and group theory are assumed. The book contains extensive references, as the author makes use of many statemants from primary sources, and a large number of figures that help the reader follow the exposition. On the whole, the book is well organized. ..."

James W. Anderson, Mathematical Reviews, Clippings from Issue 2004i

"Almost coincident with the 1985 publication of the first edition of Gerard Burde and Heiner Zieschang's otherwise magisterial Knots, V. Jones's revolutionary "polynomial invariants" blew that subject wide open. History now repeats itself. ... Just as Matveev ... provides astate-of-the-art fast track into all the mathematics surrounding the notorious Poincaré ... conjecture, G. Perelman appears to have proved not only the Poincaré conjecture, but the whole Thurston geometrization conjecture, holy grail of three-dimensional topology. ... Summing up: Recommended. Upper-division undergraduates through faculty."

D.V. Feldman, Choice - Current Reviews for College Libraries 2003

"This book provides a comprehensive and detailed account of different topics in algorithmic 3-dimensional topology … including the up-to-date results in computer enumeration of 3-manifolds. … Concerning the style the author succeeded to conform to the ground rules … as for completeness, modularity and clarity of the text. … the reviewer subscribes to the quotation from the back cover: ‘the book fills a gap in the existing literature and will become a standard reference for algorithmic 3-dimensional topology both for graduate students and researchers.’"

(Riccardo Piergallini, Zentralblatt MATH, Vol. 1048, 2004)

"The purpose of this book is to present a detailed overview of the algorithmic aspects of 3-manifold topology. … The book contains extensive references, as the author makes use of many statements from primary sources and a large number of figures that help the reader follow the exposition. On the whole, the book is well organized."

(James W. Anderson, Mathematical Reviews, 2004 i)

• Chelyabinsk State University, Chelyabinsk, Russia

  Sergei Matveev

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