Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry / Edition 1

Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry / Edition 1

by A. Bj?rner, G.M. Ziegler, Jiri Matousek
     
 

ISBN-10: 3540003622

ISBN-13: 9783540003625

Pub. Date: 06/04/2003

Publisher: Springer Berlin Heidelberg

A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. They are scattered in research papers or outlined in surveys, and they often use topological notions not commonly known among combinatorialists

Overview

A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. They are scattered in research papers or outlined in surveys, and they often use topological notions not commonly known among combinatorialists or computer scientists. This book is the first textbook treatment of a significant part of such results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level (for example, homology theory and homotopy groups are completely avoided). No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained. At the same time, many substantial combinatorial results are covered, sometimes with some of the most important results, such as Kneser's conjecture, showing them from various points of view. The history of the presented material, references, related results, and more advanced methods are surveyed in separate subsections. The text is accompanied by numerous exercises, of varying difficulty. Many of the exercises actually outline additional results that did not fit in the main text. The book is richly illustrated, and it has a detailed index and an extensive bibliography. This text started with a one-semester graduate course the author taught in fall 1993 in Prague. The transcripts of the lectures by the participants served as a basis of the first version. Some years later, a course partially based on that text was taught by Guenter M. Ziegler in Berlin. The book is based on a thoroughly rewritten version prepared during a pre-doctoral course the author taught at the ETH Zurich in fall 2001. Most of the material was covered in the course: Chapter 1 was as

Product Details

ISBN-13:
9783540003625
Publisher:
Springer Berlin Heidelberg
Publication date:
06/04/2003
Series:
Universitext Series
Edition description:
1st ed. 2003. Corr. 2nd printing 3rd Printing.
Pages:
214
Product dimensions:
6.16(w) x 9.24(h) x 0.52(d)

Table of Contents

Simplicial Complexes.- The Borsuk–Ulam Theorem.- Direct Applications of Borsuk–Ulam.- A Topological Interlude.-—2-Maps and Nonembeddability.- Multiple Points of Coincidence.

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