I am looking for books or other sources in differential topology that include topics like: vector bundles, fibration, cobordism, and other related topics. In general, if anyone has recommendation of books of advanced differential topology I would like to hear (I've already read Bott&TU, Warner). Thanks!
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14$\begingroup$ At this point, it might be better for you to start reading papers on specific topics in differential topology that you want to study more deeply. $\endgroup$– Deane YangCommented Jun 1, 2021 at 15:22
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8 Answers
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"Morse theory" by Milnor.
"Lectures on the h-cobordism theorem" by Milnor
"Characteristic classes" by Milnor and Stasheff.
"Topological methods in algebraic geometry" by Hirzebruch
"Fiber bundles" Husemoller (not differential topology per se, but discusses many topics in your question)
answered Jun 1, 2021 at 14:44
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Differential Manifolds by Antoni Kosinski covers many of these topics. It also includes a very well-written proof of the h-cobordism theorem, which I found really illuminating.
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1$\begingroup$ +1 I know this is an old question now, but if people are still referring to it, I want to second Kosinski as an advanced textbook. I think it'll make a great compliment to Wall's book I recommended above. The Dover reprint, while not quite as comprehensive as Wall, is quite inexpensive. It also contains a somewhat more geometric and visual approach then Wall, along with an impressive bibligraphy . So it compliments it very nicely. $\endgroup$ Commented Jul 28 at 23:18
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Morris Hirsch, Differential Topology
answered Jun 1, 2021 at 15:01
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I highly recommend an amazing and highly underestimated trilogy Modern Geometry. It covers not only differential geometry, but also differential and algebraic topology of manifolds.
Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. Modern geometry—methods and applications. Part I. The geometry of surfaces, transformation groups, and fields. Second edition. Translated from the Russian by Robert G. Burns. Graduate Texts in Mathematics, 93. Springer-Verlag, New York, 1992.
Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. Modern geometry—methods and applications. Part II. The geometry and topology of manifolds. Translated from the Russian by Robert G. Burns. Graduate Texts in Mathematics, 104. Springer-Verlag, New York, 1985.
Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. Modern geometry—methods and applications. Part III. Introduction to homology theory. Translated from the Russian by Robert G. Burns. Graduate Texts in Mathematics, 124. Springer-Verlag, New York, 1990.
answered Jun 4, 2021 at 23:55
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I think I got exactly what you're looking for and I'm surprised no one's suggested it yet. But then, the book is pretty new and may not be widely known outside the experts yet.
C.T. Wall-one of the founders of the subject-has finally published a polished book version his lecture notes on advanced differential topology that he gave to PhD students at Cambridge for over 4 decades. It's simply called Differential Topology and it's been published last year in hardcover by Cambridge University Press. The book covers the totality of the subject, bringing the reader from the definition of an abstract differential manifold all the way to a complete overview of the current state of equivariant cobordism. Wall doesn't explicitly say so in the book, but it's clear he expects students to have some exposure to differentiable manifolds before this as well as a good course in algebraic topology.
This isn't really a textbook. There are no explicit exercises.But let's face it-once you get past first year graduate level, you're not really talking about textbooks per se anymore. You're talking about monographs for working researchers and of course papers. That's the audience Wall is writing for and he does an exceptional job of doing it.
I think unless you want to go back to the original papers and dig up all the results from scratch yourself-something I hope you're not enough of a sadist to want to do-this is probably exactly what you're looking for.
answered Jul 23, 2021 at 5:52
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As Deane Yang suggests, you could narrow down your task by picking a short paper and digging deep enough to understand all the ingredients that go into it. For this purpose, I can highly recommend Milnor's 7-page paper where he constructs an exotic 7-sphere.
https://www.jstor.org/stable/1969983
It assumes you know bundles, characteristic classes and cobordism, but it gives you:
- Precise statements in these topics about which you can go away and find out more.
- References for where to look for more.
- Motivation for why you want to know these specific things (to understand this fantastic construction).
- And it's only 7 pages long. Once you have learned the ingredients, it's not hard to follow.
A book I like a lot for vector bundles is Atiyah's "K-theory". It very quickly explains what a complex vector bundle is and how they're classified by maps into the classifying space. It also works over compact Hausdorff spaces rather than manifolds, which means you don't go away thinking bundles are really part of smooth topology.
answered Jun 5, 2021 at 18:01
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Avoiding suggestions already made:
Guillemin and Pollack, Differential Topology, is a classic.
You can also find pieces of a lot of these things in books that are a bit broader, for example:
Topology and Geometry by Glen Bredon Lecture Notes in Algebraic Topology by Davis and Kirk
And many books on differential geometry include some of this. Notably Volume I of Spivak's Comprehensive Introduction to Differential Geometry is all differential topology.
Caveat: possibly none of these will be as advanced as you're looking for, but they all might have some pieces you're interested in.
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Let me mention a recent book:
Differential Cohomology: Categories, Characteristic Classes, and Connections, edited by Araminta Amabel, Arun Debray and Peter J. Haine, available on arXiv.
It is assembled from talks in a graduate student seminar. Although most of content is classical, the treatment is relatively modern, using $\infty$-categories, $\infty$-topoi and methods in motivic cohomology à la Voevodsky.
