Elements Of Algebraic Topology PDF

Preface

This book is intended as a text for a first-year graduate course in algebraic topology; it presents the basic material of homology and cohomology theory. For students who will go on in topology, differential geometry, Lie groups, or homological algebra, the subject is a prerequisite for later work. For other students, it should be part of their general background, along with algebra and real and complex analysis.

Geometric motivation and applications are stressed throughout. The abstract aspects of the subject are introduced gradually, after the groundwork has been laid with specific examples.

The book begins with a treatment of the simplicial homology groups, the most concrete of the homology theories. After a proof of their topological invariance and verification of the Eilenberg-Steenrod axioms, the singular homology groups are introduced as their natural generalization. CW complexes appear as a useful computational tool. This basic "core" material is rounded out with a treatment of cohomology groups and the cohomology ring.

There are two additional chapters. The first deals with homological algebra, including the universal coefficient theorems and the Kiinneth theorem. The second deals with manifolds—specifically, the duality theorems associated with the names of Poincare, Lefschetz, Alexander, and Pontryagin. tech cohomology is introduced to study the last of these.

The book does not treat homotopy theory; to do so would have made it unwieldy. There is a thorough and readable elementary treatment of the fundamental group in Massey's book [Ma]; for general homotopy theory, the reader may consult the excellent treatise by Whitehead, for which the present text is useful preparation [Wh].

Prerequisites

We assume the student has some background in both general topology and algebra. In topology, we assume familiarity with continuous functions and compactness and connectedness in general topological spaces, along with the separation axioms up through the Tietze extension theorem for normal spaces. Students without this background should be prepared to do some independent study; any standard book in topology will suffice ([D], [W], [Mu], [K], for example). Even with this background, the student might not know enough about quotient spaces for our purposes; therefore, we review this topic when the need arises (§20 and §37).  

As far as algebra is concerned, a course dealing with groups, factor groups, and homomorphisms, along with basic facts about rings, fields, and vector spaces, will suffice. No particularly deep theorems will be needed. We review the basic results as needed, dealing with direct sums and direct products in §5 and proving the fundamental theorem of finitely generated abelian groups in §11.

How the book is organized

Everyone who teaches a course in algebraic topology has a different opinion regarding the appropriate choice of topics. I have attempted to organize the book as flexibly as possible, to enable the instructor to follow his or her own preferences in this matter. The first six chapters cover the basic "core" material mentioned earlier. Certain sections marked with asterisks are not part of the basic core and can thus be omitted or postponed without loss of continuity. The last two chapters, on homological algebra and duality, respectively, are independent of one another; either or both may be covered.

The instructor who wishes to do so can abbreviate the treatment of simplicial homology by omitting Chapter 2. With this approach the topological invariance of the simplicial homology groups is proved, not directly via simplicial approximations as in Chapter 2, but as a consequence of the isomorphism between simplicial and singular theory (§34).  

When the book is used for a two-semester course, one can reasonably expect to cover it in its entirety. This is the plan I usually follow when I teach the first-year graduate course at MIT; this allows enough time to treat the exercises thoroughly. The exercises themselves vary from routine to challenging. The more difficult ones are marked with asterisks, but none is unreasonably hard.

If the book is to be used for a one-semester course, some choices will have to be made about what material to cover. One possible syllabus consists of the first four chapters in their entirety. Another consists of the first five chapters with most or all asterisked sections omitted.

A third possible syllabus, which omits Chapter 2, consists of the following:

Chapter 1

Chapter 3 (omit §27)

Chapter 4 (insert §15 before §31 and §20 before §37)

Chapters 5 and 6

If time allows, the instructor can include material from Chapter 7 or the first four sections of Chapter 8. (The later sections of Chapter 8 depend on omitted material.)

Acknowledgments

Anyone who teaches algebraic topology has had many occasions to refer to the classic books by Hilton and Wylie [H-W] and by Spanier [S]. I am no exception; certainly the reader will recognize their influence throughout the present text. I learned about CW complexes from George Whitehead; the treatment of duality in manifolds is based on lectures by Norman Steenrod. From my students at MIT, I learned what I know about motivation of definitions, order of topics, pace of presentation, and suitability of exercises.

To Miss Viola Wiley go my thanks for typing the original set of lecture notes on which the book is based.

Finally, I recall my debt to my parents, who always encouraged me to follow my own path, though it led far from where it began. To them, with love and remembrance, this book is dedicated.

Author: James R. Munkres