Free Ebook Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics), by I.M. Singer, J.A. Thorpe
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Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics), by I.M. Singer, J.A. Thorpe
Free Ebook Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics), by I.M. Singer, J.A. Thorpe
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At the present time, the average undergraduate mathematics major finds mathematics heavily compartmentalized. After the calculus, he takes a course in analysis and a course in algebra. Depending upon his interests (or those of his department), he takes courses in special topics. Ifhe is exposed to topology, it is usually straightforward point set topology; if he is exposed to geom etry, it is usually classical differential geometry. The exciting revelations that there is some unity in mathematics, that fields overlap, that techniques of one field have applications in another, are denied the undergraduate. He must wait until he is well into graduate work to see interconnections, presumably because earlier he doesn't know enough. These notes are an attempt to break up this compartmentalization, at least in topology-geometry. What the student has learned in algebra and advanced calculus are used to prove some fairly deep results relating geometry, topol ogy, and group theory. (De Rham's theorem, the Gauss-Bonnet theorem for surfaces, the functorial relation of fundamental group to covering space, and surfaces of constant curvature as homogeneous spaces are the most note worthy examples.) In the first two chapters the bare essentials of elementary point set topology are set forth with some hint ofthe subject's application to functional analysis.
- Sales Rank: #486142 in Books
- Published on: 1976-12-10
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x .63" w x 6.14" l, 1.10 pounds
- Binding: Hardcover
- 232 pages
Most helpful customer reviews
7 of 7 people found the following review helpful.
Nice overview of Point-Set, Algebraic, and Differential Topology
By Michael C. Wilson
I believe that one of the authors, I. Singer is a Fields Medalist, so this is co-written by one of the masters in the subject. It is one of the few books I'm aware of that covers point-set, algebraic, and differential topology. However, this is not in any way an exhaustive text. It is very spare and to the point. This is a small book packed with information. No exercises are included. It is not particularly advance in any one area of topology, so keep in mind that this is just an overview of all the main areas. However, just because it is not particularly advanced, does not mean it is easy to read for someone completely unfamiliar with the material. I tried reading the early chapters as a first introduction to topology, and didn't know what was going on. Reading of it got better when I read it side-by-side with the books by Munkres and Armstrong. This book is pretty abstract and dense, so it will read slowly. A lot of the finer details in the subjects are found in other texts.
16 of 21 people found the following review helpful.
Concise and modern
By A Customer
First, a small carp at Amazon: The book has two authors and since John Thorpe is the junior and by far the less famous, I suspect that he actually did most of the writing. Please give him his due!
This is a very dense book. While this makes for rough sledding for the first timer, it's also an exciting introduction to modern topology and geometry and a good first step for those interested in such things in physics as gauge theories and superstrings. It's worth the effort.
Starting with the basics of set theory, the first couple chapters take the reader through point set topology. The next couple chapters introduce algebraic topology. The rest of the book is about the algebraic topology of differentiable manifolds and a very clean, modern introduction to the classical differential geometry of surfaces. The only caveat is, as Spivak says, "a weird proof of the de Rham theorem" in Chapter 6. I'm torn about this. The proof in Warner's "Foundations of Differentiable Manifold and Lie Groups" is much cleaner and better lends itself to other applications, but involves lots of machinery. The proof in Singer and Thorpe is a lot less elegant, using the lowest level tools possible. This makes the learning curve shorter and may make the theorem more clear, but may also obscure the big picture. Much of the important work in algebraic topology over the next 20 years and theoretical physics up to now is related to this result. Though much of this work was developed by Singer with his collaborater Michael Atiyah, their approach is closer to Warner's than to that in Singer and Thorpe.
For any particular topic in this book, you can find sources that you'll undoubtably find more digestible. This is the only book that brings them all together. It's an audacious effort.
2 of 2 people found the following review helpful.
A useful monograph for topology
By Stan Sholar
Although ostensibly for undergraduates, this monograph text requires considerable ability in abstract thinking. A well developed capability and sophistication with certain prerequisite math skills is useful. The text does, however, accomplish its claim, namely providing a shortcut pathway toward some of the more fun stuff in advanced mathematics, like deRham cohomology theory, without requiring a long chain of prerequisite courses, usually amounting to an undergraduate major in mathematics. For students who have a natural affinity for advanced mathematics, this approach could provide additional motivation to pursue those interests. For non-math majors with a particular interest in cohomology theory, this shortcut works amazingly well.
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