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Homology Theory: An Introduction to Algebraic Topology (Graduate Texts in Mathematics), by James W. Vick
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This introduction to some basic ideas in algebraic topology is devoted to the foundations and applications of homology theory. After the essentials of singular homology and some important applications are given, successive topics covered include attaching spaces, finite CW complexes, cohomology products, manifolds, Poincare duality, and fixed point theory. This second edition includes a chapter on covering spaces and many new exercises.
- Sales Rank: #1843468 in eBooks
- Published on: 2012-12-21
- Released on: 1973-05-31
- Format: Kindle eBook
From the Back Cover
This book is designed to be an introduction to some of the basic ideas in the field of algebraic topology. In particular, it is devoted to the foundations and applications of homology theory. The only prerequisite for the student is a basic knowledge of abelian groups and point set topology. The essentials of singular homology are given in the first chapter, along with some of the most important applications. In this way the student can quickly see the importance of the material. The successive topics include attaching spaces, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology products, manifolds, Poincare duality, and fixed point theory. Throughout the book, the approach is as illustrative as possible, with numerous examples and diagrams. Extremes of generality are sacrificed when they are likely to obscure the essential concepts involved. The book is intended to be easily read by students as a textbook for a course or as a source for individual study. This second edition has been expanded to include a new chapter on covering spaces, as well as additional illuminating exercises. The conceptual approach is again used to show how lifting problems give rise to the fundamental group and its properties.
Most helpful customer reviews
0 of 1 people found the following review helpful.
Five Stars
By Amazon Customer
I didn't read the book yet.but generally it looks great
0 of 1 people found the following review helpful.
Three Stars
By Vitaly Zaderman
good
2 of 2 people found the following review helpful.
Very simply, the best introductory book on homology there is.
By Joseph
I've studied and refreshed my understanding of algebraic topology from lots of books over the years, and I always come back to this one. It's the best of the lot by far, for me -- easy to read, beautifully geometrically motivated, and displaying as light a touch as possible with algebraic and categorical details (which are introduced in digestible pieces as the need for them arises). I'd say that anyone with a good facility for basic group theory, even with a quite minimal background in point-set topology, can come to understand homology theory quite easily by following this book. I love the fact that Vick takes the reader straight into the thick of things, quickly getting to computations of homology groups of spheres and deducing a slew of nice geometric theorems from them (invariance of dimension, invariance of domain, Brouwer's fixed point theorem, the "hairy ball" theorem, the Jordan separation theorem, etc.) -- and that he doesn't waste any time with the tedium of simplicial complexes, the simplicial approximation theorem, and all that. It has always seemed perverse to me that beginning books tend to start with the simplicial theory; true, it's conceptually very basic, but its technical details are very easy to get bogged down in. Vick begins with singular homology, which is far easier to set up on a technical level than its simplicial cousin. After a number of very worthwhile deductions from the singular theory, he introduces CW complexes and shows that their homology is the same as that obtained from the singular theory. His treatments of products and Poincare duality are also outstanding. The best thing about this book is that it's genuinely accessible to the beginning graduate student. You don't need to have had a course in category theory to read and understand this book. (In fact, seeing some things in a concrete setting here, like direct limits, will undoubtedly help people understand their more abstract categorical versions later on in graduate school). I will make one criticism of this book's way into homology theory, and that is that Vick neglects to prove Hurewicz's result that the first homology group of a path-connected space is the abelianization of the fundamental group. Although this result is nowhere needed in the book, to me it's really essential for understanding the geometric content of the integer coefficient of a given singular n-simplex within a given n-chain, at least in the easily visualized case n = 1. (It also helps in solving one of the exercises in this book, where one has to construct a map from the n-sphere to itself with any given integer as its degree; once you understand the Hurewicz result, it's easy to find such a map on the circle in the complex plane, and then repeatedly iterate the suspension operator to increase the dimension of the sphere.) Apart from this detail, I love everything about this book. There are some typos, but that's true in most books, and they didn't bother me that much. I recommend this book to any novice student of algebraic topology, and especially to any who feel that their background in abstract algebraic machinery (category theory, homological algebra) is somewhat lacking. You'll learn it well from this book. Enjoy it.
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