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Fred H. Croom

Basic Concepts of
Algebraic Topology


New York



Fred H. Croom
The University of the South
Sewanee, Tennessee 37375

Editorial Board
F. W. Gehring

P. R. Halmos

University of Michigan
Department of Mathematics
Ann Arbor, Michigan 48104

University of California,
Department of Mathematics
Santa Barbara, California 93106

AMS Subject Classifications: 55-01

Library of Congress Cataloging in Publication Data
Croom, Fred H
1941Basic concepts of algebraic topology.
(Undergraduate texts in mathematics)
Bibliography: p.
Includes index.
1. Algebraic topology. I. Title.

All rights reserved.
No part of this book may be translated or reproduced in any form without written permission
from Springer-Verlag.
© 1978 by Springer-Verlag, New York Inc.
Printed in the United States of America.

ISBN 0-387-90288-0 Springer-Verlag New York
ISBN 3-540-90288-0 Springer-Verlag Berlin Heidelberg


This text is intended as a one semester introduction to algebraic topology
at the undergraduate and beginning graduate levels. Basically, it covers
simplicial homology theory, the fundamental group, covering spaces, the
higher homotopy groups and introductory singular homology theory.
The text follows a broad historical outline and uses the proofs of the
discoverers of the important theorems when this is consistent with the
elementary level of the course. This method of presentation is intended to
reduce the abstract nature of algebraic topology to a level that is palatable
for the beginning student and to provide motivation and cohesion that are
often lacking in abstact treatments. The text emphasizes the geometric
approach to algebraic topology and attempts to show the importance of
topological concepts by applying them to problems of geometry and
The prerequisites for this course are calculus at the sophomore level, a
one semester introduction to the theory of groups, a one semester introduction to point-set topology and some familiarity with vector spaces. Outlines
of the prerequisite material can be found in the appendices at the end of
the text. It is suggested that the reader not spend time initially working on
the appendices, but rather that he read from the beginning of the text,
referring to the appendices as his memory needs refreshing. The text is
designed for use by college juniors of normal intelligence and does not
require "mathematical maturity" beyond...