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Algebraic Topology, by Edwin H. Spanier
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This book surveys the fundamental ideas of algebraic topology. The first part covers the fundamental group, its definition and application in the study of covering spaces. The second part turns to homology theory including cohomology, cup products, cohomology operations and topological manifolds. The final part is devoted to Homotropy theory, including basic facts about homotropy groups and applications to obstruction theory.
- Sales Rank: #1535481 in Books
- Brand: Brand: Springer
- Published on: 2008-05-23
- Original language: English
- Number of items: 1
- Dimensions: 9.25" h x 1.24" w x 6.10" l, 1.69 pounds
- Binding: Paperback
- 548 pages
- Used Book in Good Condition
From the Back Cover
The reader of this book is assumed to have a grasp of the elementary concepts of set theory, general topology, and algebra. Following are brief summaries of some concepts and results in these areas which are used in this book. Those listed explicitly are done so either because they may not be exactly standard or because they are of particular importance in the subsequent text.
Most helpful customer reviews
13 of 14 people found the following review helpful.
Pioneering text
By topoman
This book was an incredible step forward when it was written (1962-1963). Lefschetz's Algebraic Topology (Colloquium Pbns. Series, Vol 27) was the main text at the time. A large number of other good to great books on the subject have appeared since then, so a review for current readers needs to address two separate issues: its suitability as a textbook and its mathematical content.
I took the course from Mr. Spanier at Berkeley a decade after the text was written. He was a fantastic teacher - one of the two best I've ever had (the other taught nonlinear circuit theory). We did NOT use this text, except as a reference and problem source. He had pretty much abandonded the extreme abstract categorical approach by then. The notes I have follow the topical pattern of the book, but are so modified as to be essentially a different book, especially after covering spaces and the first homotopy group. His statement was that his treatment had changed since the subject had changed significantly. So much more has changed since then that I would not recommend this book as a primary text these days. Bredon's Topology and Geometry (Graduate Texts in Mathematics) is much better suited to today's student.
So, why did I give it four stars? First, notice that it splits stylewise into three segments, corresponding the treatment of its material in a three quarter academic year. The first three chapters (intro, covering spaces, polyhedral) have really not been superceded in a beginning text. Topics are covered very thoroughly, aiding the student new to the subject. The next three chapters (homology) are written much with much less explanation included - indeed, some areas leave much to the reader to discover and, consequently, aren't very helpful if the instructor doesn't fill in the details (the text expects a rather rapid mathematical maturation from the first part - too much of a ramp in my opinion), but the text is comprehensive. The last section (homotopy theory, obstruction theory and spectral sequences) should just be treated as a reference - it'd be hard to find all this material in such a compact form elsewhere and the obstruction theory section has fantastic coverage of what was known as of the writing of this book. It's way too terse for a novice to learn from and there are some great books out there these days on the material.
21 of 22 people found the following review helpful.
Excellent reference, poor textbook
By A Customer
This book is terrific as a reference for those who already know the subject, but if you teach algebraic topology it would be dangerous to use it as a graduate text (unless you're willing to supplement it extensively). The basic problem is that Spanier does not teach students how to compute effectively because his abstract, high-powered algebraic approach obscures the underlying geometry, which is not developed at all. Here I'd recommend the books by Munkres, or Greenberg; even the old-fashioned treatment of Lefschetz, with its explicit and rather cumbersome treatment of cohomology, could serve as an antidote to Spanier. Somewhere, the student has to acquire a good intuitive feeling for the geometry underlying the subject (the same can be said of algebraic geometry -- here earlier work (e.g., of the Italian school, Weil's old book on intersection theory, ...) should not be neglected entirely in favor of Grothendieck et al., for something essential is lost)
That said, if you already know the subject Spanier's book is an excellent reference. Even here, though, you'll need to provide some details toward the ends of the later chapters. Each chapter starts out relatively easily and works up to a crescendo, the treatment becoming terser and more advanced.
I give it four stars (5 for mathematical quality, 3 for usefulness as a text). The first three chapters deal with covering spaces and fibrations; the middle three with (co)homology and duality; the last three with general homotopy theory, obstruction theory, and spectral sequences. Some of Serre's classical results on finiteness theorems for homotopy groups are presented.
5 of 6 people found the following review helpful.
Excellent reference, poor textbook
By A Customer
This book is terrific as a reference for those who already know the subject, but if you teach algebraic topology it would be dangerous to use it as a graduate text (unless you're willing to supplement it extensively). The basic problem is that Spanier does not teach students how to compute effectively because his abstract, high-powered algebraic approach obscures the underlying geometry, which is not developed at all. Here I'd recommend the books by Munkres, or Greenberg; even the old-fashioned treatment of Lefschetz, with its explicit and rather cumbersome treatment of cohomology, could serve as an antidote to Spanier. Somewhere, the student has to acquire a good intuitive feeling for the geometry underlying the subject (the same can be said of algebraic geometry -- here earlier work (e.g., of the Italian school, Weil's old book on intersection theory, ...) should not be neglected entirely in favor of Grothendieck et al., for something essential is lost)
That said, if you already know the subject Spanier's book is an excellent reference. Even here, though, you'll need to provide some details toward the ends of the later chapters. Each chapter starts out relatively easily and works up to a crescendo, the treatment becoming terser and more advanced.
I give it four stars (5 for mathematical quality, 3 for usefulness as a text). The first three chapters deal with covering spaces and fibrations; the middle three with (co)homology and duality; the last three with general homotopy theory, obstruction theory, and spectral sequences. Some of Serre's classical results on finiteness theorems for homotopy groups are presented.
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