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Rating: 
Summary: Concise and modern
Review: 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.
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