Using the Borsuk-Ulam Theorem - J.Matousek.pdf - Topologia ogolna - mwt3

Jirı Matousek

Using the

Borsuk–Ulam Theorem

Lectures on Topological Methods

in Combinatorics and Geometry

Written in cooperation with

Anders Bjorner and Gunter M. Ziegler

2nd, corrected printing

Jirı Matousek

Charles University

Department of Applied Mathematics

Malostransk´enam. 25

118 00 Praha 1

Czech Republic

matousek@kam.mff.cuni.cz

Corrected 2nd printing 2008

ISBN 978-3-540-00362-5

e-ISBN 978-3-540-76649-0

Universitext

Library of Congress Control Number: 2007937406

Mathematics Subject Classiﬁcation (2000): 05-01, 52-01, 55M20; 05C15, 05C10, 52A35

2003 Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the

material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations,

recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in

data banks. Duplication of this publication or parts thereof is permitted only under the pro-

visions of the German Copyright Law of September 9, 1965, in its current version, and per-

mission for use must always be obtained from Springer. Violations are liable to prosecution

under the German Copyright Law.

The use of general descriptive names, registered names, trademarks, etc. in this publication

does not imply, even in the absence of a speciﬁc statement, that such names are exempt

from the relevant protective laws and regulations and therefore free for general use.

Cover design: design&production GmbH, Heidelberg

Printed on acid-free paper

987654321

springer.com

Preface

A number of important results in combinatorics, discrete geometry, and the-

oretical computer science have been proved by surprising applications of al-

gebraic topology. Lovasz’s striking proof of Kneser’s conjecture from 1978 is

among the ﬁrst and most prominent examples, dealing with a problem about

ﬁnite sets with no apparent relation to topology.

During the last two decades, topological methods in combinatorics have

become more elaborate. On the one hand, advanced parts of algebraic topol-

ogy have been successfully applied. On the other hand, many of the earlier

results can now be proved using only fairly elementary topological notions

and tools, and while the ﬁrst topological proofs, like that of Lovasz, are mas-

terpieces of imagination and involve clever problem-speciﬁc constructions,

reasonably general recipes exist at present. For some types of problems, they

suggest how the desired result can be derived from the nonexistence of a

certain map (“test map”) between two topological spaces (the “conﬁguration

space” and the “target space”). Several standard approaches then become

available for proving the nonexistence of such a map. Still, the number of dif-

ferent combinatorial results established topologically remains relatively small.

This book aims at making elementary topological methods more easily

accessible to nonspecialists in topology. It covers a number of substantial

combinatorial and geometric results, and at the same time, it introduces

the required material from algebraic topology. Background in undergraduate

mathematics is assumed, as well as a certain mathematical maturity, but no

prior knowledge of algebraic topology. (But learning more algebraic topology

from other sources is certainly encouraged; this text is no substitute for proper

foundations in that subject.)

We concentrate on topological tools of one type, namely, the Borsuk–

Ulam theorem and similar results. We develop a systematic theory as far

as our restricted topological means su ce. Other directions of research in

topological methods, often very beautiful and exciting ones, are surveyed in

Bjorner [Bjo95].

History and notes on teaching. This text started with a course I taught in

fall 1993 in Prague (a motivation for that course is mentioned in Section 6.8).

Transcripts of the lectures made by the participants served as a basis of the

ﬁrst version. Some years later, a course partially based on that text was

Using the Borsuk-Ulam Theorem - J.Matousek.pdf

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: