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Differential Forms in Algebraic Topology (Graduate Texts in Mathematics) [Kindle Edition]

Raoul Bott , Loring W. Tu
5.0 out of 5 stars  See all reviews (12 customer reviews)

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Book Description

Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology.


Editorial Reviews

Review

“Bott and Tu give us an introduction to algebraic topology via differential forms, imbued with the spirit of a master who knew differential forms way back when, yet written from a mature point of view which draws together the separate paths traversed by de Rham theory and homotopy theory. Indeed they assume "an audience with prior exposure to algebraic or differential topology". It would be interesting to use Bott and Tu as the text for a first graduate course in algebraic topology; it would certainly be a wonderful supplement to a standard text.

“Bott and Tu write with a consistent point of view and a style which is very readable, flowing smoothly from topic to topic. Moreover, the differential forms and the general homotopy theory are well integrated so that the whole is more than the sum of its parts. "Not intended to be foundational", the book presents most key ideas, at least in sketch form, from scratch, but does not hesitate to quote as needed, without proof, major results of a technical nature, e.g., Sard's Theorem, Whitney's Embedding Theorem and the Morse Lemma on the form of a nondegenerate critical point.”

—James D. Stasheff (Bulletin of the American Mathematical Society)

 

“This book is an excellent presentation of algebraic topology via differential forms. The first chapter contains the de Rham theory, with stress on computability. Thus, the Mayer-Vietoris technique plays an important role in the exposition. The force of this technique is demonstrated by the fact that the authors at the end of this chapter arrive at a really comprehensive exposition of Poincaré duality, the Euler and Thom classes and the Thom isomorphism.

“The second chapter develops and generalizes the Mayer-Vietoris technique to obtain in a very natural way the Čech-de Rham complex and the Čech cohomology for presheaves. The third chapter on spectral sequences is the most difficult one, but also the richest one by the various applications and digressions into other topics of algebraic topology: singular homology and cohomology with integer coefficients and an important part of homotopy theory, including the Hopf invariant, the Postnikov approximation, the Whitehead tower and Serre’s theorem on the homotopy of spheres. The last chapter is devoted to a brief and comprehensive description of the Chern and Pontryagin classes.

“A book which covers such an interesting and important subject deserves some remarks on the style: On the back cover one can read “With its stress on concreteness, motivation, and readability, Differential forms in algebraic topology should be suitable for self-study.” This must not be misunderstood in the  ense that it is always easy to read the book. The authors invite the reader to understand algebraic topology by completing himself proofs and examples in the exercises. The reader who seriously follows this invitation really learns a lot of algebraic topology and mathematics in general.”

—Hansklaus Rummler (American Mathematical Society)


Product Details

  • File Size: 6676 KB
  • Print Length: 338 pages
  • Publisher: Springer; 1st ed. 1982. Corr. 3rd printing 1995 edition (April 17, 2013)
  • Sold by: Amazon Digital Services, Inc.
  • Language: English
  • ASIN: B000YFYQ6K
  • Text-to-Speech: Enabled
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  • Lending: Not Enabled
  • Amazon Best Sellers Rank: #1,457,455 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
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50 of 50 people found the following review helpful
5.0 out of 5 stars A unique mathematics book January 22, 2002
By sigfpe
Format:Hardcover
This book is almost unique among mathematics books in that it strives to ensure that you have the clearest picture possible of the topics under discussion. For example almost every text that discusses spectral sequences introduces them as a completely abstract machine that pumps out theorems in a mysterious way. But it turns out that all those maps actually have a clear meaning and Bott and Tu get right in there with clear diagrams showing exactly what those maps mean and where the generators of the various groups get mapped. It's clear enough that you can almost reach out and touch the things :-) And the same is true of all of the other constructions in the book - you always have a concrete example in mind with which to test out your understanding.
That makes this one of my all time favourite mathematics texts.
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41 of 41 people found the following review helpful
Format:Hardcover
Wonderful book. The real theme of this book is to get the reader to some powerful and compelling applications of algebraic topology and comfort with spectral sequences. The use of differential forms avoids the painful and for the beginner unmotivated homological algebra in algebraic topology. E.g., For example, the wedge product of differential forms allow immediate construction of cup products without digression into acyclic models, simplicial sets, or Eilenberg-Zilber theorem. The authors later come back and do the now-motivated version for singular homology later.
I really like the idea of using spectral sequences from the beginning. It quickly brings in the actual flavor of algebraic topology by introducing today's workhorse tool. Also, because spectral sequences take a long time to become second nature (at least they did for me), the earlier the exposure to them the quicker the reader will be able to do more advanced topics comfortably. Again, spectral sequences are introduced in a painless special case, that of a double complex, and more difficult cases are not treated until the reader feels comfortable with basic spectral sequence calculations.
Finally, the prerequisites are less than or equal to those for other algebraic topology books, making this a nice choice for a first exposure to algebraic topology.
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24 of 24 people found the following review helpful
5.0 out of 5 stars A first class job August 17, 2001
Format:Hardcover
The authors of this book, through clever examples and in-depth discussion, give the reader a rare accounting of some of the important concepts of algebraic topology. The introduction motivates the subject nicely, and the authors succeed in giving the reader an appreciation of where the concepts of algebraic topology come from, how they do their jobs, and their limitations. The de Rham cohomology, which is the main subject of the book, is explained in here in a way that gives the reader an intuitive and geometric understanding, which is sorely needed, especially for physicists who are interested in applications. As an example, they give a neat argument as to why de Rham cohomology cannot detect torsion.
In chapter 1, the authors get down to the task of constructing de Rham cohomology, starting with the de Rham complex on R(n). The de Rham complex is then specialized to the case where only C-infinity functions with compact support are used, giving the de Rham complex with compact supports on R(n). The de Rham complex is then generalized to any differentiable manifold and the de Rham cohomology computed using the Mayer-Vietoris sequence.
The discussion gets a little more involved when the authors characterize the cohomology of a fiber bundle. The all-important Thom isomorphism for vector bundles, is treated in detail. The authors give several good examples of the Poincare duals of submanifolds. The connection to ideas in differential topology is readily apparent in this chapter, namely transversality and the degree of a map. In addition, the first construction of a characteristic class, the Euler class, is done in this chapter.
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12 of 12 people found the following review helpful
5.0 out of 5 stars wonderfully clear, useful book February 6, 2006
By twit
Format:Hardcover
I agree with the other reviews, and only wanted to add to one of them that in regard to examples of chern classes, I believe they also use the whitney formula to derive the chern classes of a hypersurface from that of projective space, which really expands the realm of examples significantly.

This was all I needed in writing my notes on the Riemann Roch theorem for hypersurfaces in 3 and 4 space, for instance. I felt I knew little about concrete chern classes, but I was able to take the presentation in this book and use it for my purposes immediately.
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10 of 10 people found the following review helpful
5.0 out of 5 stars a masterpiece of exposition December 12, 2007
Format:Hardcover
This is a beautiful book which I have read and re-read with much profit and pleasure over the years. It presents topics in a very unusual order, which minimizes boring technicalities and develops intuition. Everything is very concrete and explicit, with lots of nice pictures and diagrams.

The book begins with a clear and concise treatment of deRham cohomology. If one hasn't seen differential forms before, then it might be a bit too brief and one might need to supplement it. But if one is comfortable with differential forms, then de Rham theory is a setting in which theorems such as Poincare duality can be proved with a minimum of pain. It is also very edifying to see the Poincare dual of a submanifold as a differential form. There is then a natural transition to Cech cohomology and double complexes. With this as a warmup, it is then a small additional step to spectral sequences (although the derived couple approach used here is perhaps not the most elementary possible). This machinery is then used to discuss an assortment of topics in homotopy theory and characteristic classes, which always sticks to the most important points without getting bogged down in technicalities.

It is highly unusual that the definition of singular homology only comes after the introduction of spectral sequences! This book might be best appreciated if one has some familiarity with singular homology and wants to better understand its geometric meaning.

Despite the avoidance of technicalities, the book is carefully written, although there is the occasional sign error. For example, the sign given for the Lefschetz fixed point theorem is wrong for odd-dimensional manifolds; try it for the circle and you will see. (Several other books make the same mistake.)
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