Lectures on Algebraic Topology
This is essentially a book on singular homology and cohomology with special emphasis on products and manifolds. It does not treat homotopy theory except for some basic notions, some examples, and some applica tions of (co-)homology to homotopy. Nor does
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H erausgegeben von 1. L. Doo b
A. Grothendieck E. Heinz F. Hirzebruch E. Hopf W. Maak S. Mac Lane W.Magnus J.K.Moser M. M. Postnikov F. K. Schmidt D. S. Scott K. Stein
Geschaftsfuhrende Herausgeber B. Eckmann und B. L. van der Waerden
A. Dold
Lectures on Aigebraic Topology
With 10 Figures
Springer-Verlag Berlin IIcidc1bcrg GmbH
Albrecht Dold Mathematisches Institut der Universität Heidelberg
Geschäftsführende Herausgeber
B. Eckmann Eidgenössische Technische Hochschule Zürich B. L. van der Waerden Mathematisches Institut der Universität Zürich
AMS Subject Classifications (1970) Primary 57A65, 55BI0, 55B45, 55JI5-25, 54E60 Secondary 55C20-25
ISBN 978-3-662-00758-7 ISBN 978-3-662-00756-3 (eBook) DOI 10.1007/978-3-662-00756-3
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© by Springer-Verlag Berlin Heidelberg 1972. Originally published by Springer-Verlag Berlin Heidelberg New Y ork 1972 Softcover reprint of the hardcover 1st edition 1972 Library of Congress Catalog Card Number 72-79062
ToZ.
Foreword
This is essentially a book on singular homology and cohomology with special emphasis on products and manifolds. It does not treat homotopy theory except for some basic notions, some examples, and some applications of (co-)homology to homotopy. Nor does it deal with general(-ised) homology, but many formulations and arguments on singular homology are so chosen that they also apply to general homology. Because of these absences I have also omitted spectral sequences, their main applications in topology being to homotopy and general (co-)homology theory. Cechcohomology is treated in a simple ad hoc fashion for locally compact subsets of manifolds; a short systematic treatment for arbitrary spaces, emphasizing the universal property of the Cech-procedure, is contained in an appendix. The book grew out of a one-year's course on algebraic topology, and it can serve as a text for such a course. For a shorter basic course, say of half a year, one might use chapters II, III, IV (§§ 1-4), V (§§ 1-5, 7, 8), VI (§§ 3, 7, 9, 11, 12). As prerequisites the student should know the elementary parts of general topology, abelian group theory, and the language of categories - although our chapter I provides a little help with the latter two. For pedagogical reasons, I have treated integral homology only up to chapter VI; if a reader or teacher prefers to have general coefficients from the beginning he needs to make only minor adaptions. As to the outlay of the book, there are eight chapters, I-VIII, and an appendix, A; each of these is subdivided into several sections, § 1, 2, ..... Definitions, pro
