E-Book Overview
The book is a continuation of the previous book by the author (Elements of Combinatorial and Differential Topology, Graduate Studies in Mathematics, Volume 74, American Mathematical Society, 2006). It starts with the definition of simplicial homology and cohomology, with many examples and applications. Then the Kolmogorov-Alexander multiplication in cohomology is introduced. A significant part of the book is devoted to applications of simplicial homology and cohomology to obstruction theory, in particular, to characteristic classes of vector bundles. The later chapters are concerned with singular homology and cohomology, and Cech and de Rham cohomology. The book ends with various applications of homology to the topology of manifolds, some of which might be of interest to experts in the area. The book contains many problems; almost all of them are provided with hints or complete solutions.
E-Book Content
Elements of Homology Theory
Elements of Homology Theo ry v. V. Prasolov
Graduate Studies in Mathematics Volume 81
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American Mathematical Society Providence, Rhode Island
David Cox (Chair) Walter Craig
N. V. Ivanov Steven G. Krantz
B. B. IIpaconoB 8JIEMEHThI TEOPI1I1 rOMOJIOrl1H MIIHMO, MocKBa, 2005 This work was originally published in Russian by MIIHMO under the title "8neMeHTLI TeOpH:" rOMOnOr"�"
©
2005. The present translation was created under license for the
American Mathematical Society and is published by p ermission. Translated from the Russian by Olga Sipacheva
2000
Mathematics Subject ClasSIfication.
Primary 55-01.
For additional information and updates on this book, visit www
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Library of Congress Cataloging-in-Publication Data Prasolov, V. V. (Viktor Vasil'evich)
[Elementy teorii gomologii. English]
Elements of homology theory / V. V. Prasolov. p. cm. - (Graduate studies in mathematics; v. 81) Includes bibliographical references and index. ISBN-13: 978-0-8218-3812-9 (alk. paper) ISBN-I0: 0-8218-3812-1 (alk. paper) 1. Homology theory. I. Title. QA612.3.P73
2007
514'.23---dc22
2006047074
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2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.
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Contents
Preface
vii
Notation Chapter 1.
ix
Simplicial Homology
1
§1.
Definition and Some Properties
1
§2.
Invariance of Homology
6
§3.
Relative Homology
12
§4.
Cohomology and Universal Coefficie