Differential Forms in Algebraic Topology

Graduate Texts in Mathematics #82

by Raoul Bott, Loring W. Tu

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.

Genres: Mathematics, Geometry, Nonfiction, Science, Textbooks

490 pages, Kindle Edition

First published May 24, 1982

Reviews

[R · Rating 5 out of 5]

A magnificently frustrating book. Or frustratingly magnificent, depending on the day.

[Sam · Rating 5 out of 5]

I wish this were a standard text for learning about smooth manifolds though it would be quite a bit of effort for those completely unfamiliar with smooth manifolds. One is able to maintain some intuition while also rigorously understanding algebraic topological invariants for manifolds. I learned a good deal from this book about the Thom isomorphism and also spectral sequences. It contains my favorite example of the spectral sequence for computing the homology of loop spaces of spheres.

[psb · Rating 5 out of 5]

Cohomology is making a come back at the Downtown Clubs, so time to review de Rham Theorem etc.

Hard (for me) but Stunning ... this is what math is all about ... but I'm willing (and able) to put more effort into geometry/topology than say algebra or number theory.

Heh: "Editions before 1995 should be discarded." --Loring Tu

[Fan Wang · Rating 5 out of 5]

Cannot think of any other maths books better than this one.

[Steve Dalton · Rating 5 out of 5]

Superb. The material is presented in the clearest and cleanest way possible, with certain examples brought up again and again to drive home important points. And the treatment of spectral sequences is fantastic!

The material in here is probably standard fare for graduate students, but I think the real benefit of reading this book is the way it gets you to think about differential forms, cohomology, fiber bundles, long exact sequences, characteristic classes, Poincare duality, .... If I had to complain, I would say that a few more pictures would've been nice. But honestly, the explanations are so easy-to-follow, not many pictures are needed!

I just read another reviewer's post, and he/she mentioned the de Rham - Cech double complex. I almost forgot the masterful treatment of this wonderful object! It comes up continuously, and the amount of results and ideas Bott and Tu manage to wring from it is mindblowing (and enlightening).

[Jon Paprocki · Rating 5 out of 5]

I haven't read the entire thing yet, but I've digested at least half of it so far and this is truly one of the best textbooks I've read. It has shed new light on topics that I thought I knew through and through, and finally made crystal clear some ideas that haven't quite sat right with me for years. I have yet to find a proof in this book that was shoddily written. I was especially impressed by the very sexy and sleek proof on the isomorphism between de Rham cohomology and the Cech cohomology of the constant real presheaf, as I am accustomed to finding proofs of isomorphisms between cohomology theories to be clunky. Also, I will forever know the de Rham complex as a God-given set of differential equations.