I am interested in differential cohomology & secondary characteristic classes and am currently studying the notes by Ulrich Bunke. While these are nice notes, I sometimes find it hard to fill in the gaps in the proofs. Could someone please suggest a reference that I could use to supplement the article ? Are there any other good references on this subject for a beginner ?

As regards my background, I have studied homology and cohomology theory, basic homotopy theory & topology of fibre bundles (from Husemoller's book) and differential geometry (connections, curvature, deRham cohomology, chern classes).

Thanks a lot !

  • $\begingroup$ If this doesn't get a response here after the bounty runs out, you might want to try MathOverflow, because this appears to be fairly high-level. $\endgroup$ – Chill2Macht Oct 14 '16 at 19:59
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    $\begingroup$ Are you familiar with basic Chern-Weil theory? (Not sure if that's what you meant by including Chern classes in the differential geometry section.) I've only skimmed the paper you linked to, but that seems like the best place to start. $\endgroup$ – anomaly Oct 14 '16 at 20:05
  • $\begingroup$ @anomaly Yes, I am aware of Chern-Weil theory and that is what I meant. $\endgroup$ – user90041 Oct 22 '16 at 16:57
  • $\begingroup$ In that case, could you be more specific about what you have in mind? I would point you toward different references if you were looking for more background in the paper's treatment of fiber bundles (e.g., Chern classes in general, the Leray-Hirsch theorem, classifying spaces, etc.), secondary cohomology operations, K-theory, etc. $\endgroup$ – anomaly Oct 22 '16 at 18:10
  • $\begingroup$ @anomaly Actually I have studied Husemoller's book on Fibre bundles so I already know Chern classes, Leray-Hirsch theorem and basic Topological K-theory. However I do not know much about secondary cohomology operations though and I have never studied 'Bordism' and Thom spectra MSO_n previously (which the paper uses in section 2.6). It would be nice of you to please suggest some references for these. And if you could advise me some alternate references for differential cohomology and differential characters that I could supplement, that would be very helpful $\endgroup$ – user90041 Oct 22 '16 at 18:20

Here are some references:

  • May's "Concise Course in Algebraic Topology" is an odd book: It's a treatment of a standard first course in algebraic topology (e.g., Hatcher) under the assumption that you're already familiar with everything in it. That is, it explains what's really going using the proper machinery from algebraic topology and commutative algebra that programs defer to later courses. The first third is general topology (homotopy lifting, etc.), along the lines of what the first part of Spanier covers. The second third is the algebraic part of 'algebraic topology', covering homology and cohomology. The last third is the relevant bit: K-theory, cobordism, and more with characteristic classes than you'd find in Milnor and Stasheff.
  • Chapter 15 ("The Technical Chapter") of Berger's "Panoramic View of Riemannian Geometry" seems to cover the sort of material in the paper. I'm going off the table of contents for that chapter, though; I haven't read it myself, or at least haven't looked it at since grad school. (It's a massive book, almost a thousand pages.)
  • Bott and Tu's "Differential Forms in Algebraic Topology" is less focused on the smooth case than the title implies, but it may be useful to you. (It's also quite possible that the smooth bits will already be familiar to you and the more general bits--- e.g., Postnikov towers--- may not be quite what you're looking for.) It's also the best-written math textbook I've come across.
  • Hatcher's "Vector Bundles and K-Theory" is more algebraic in flavor than you're probably looking for, but it's conveniently available for free online.
  • Hirzebruch's "Topological Methods in Algebraic Geometry" looks interesting and relevant here, but I haven't read too much of it myself. (It's not as well written as the other books here, and I happen to dislike algebraic geometry.) Specifically, it covers some topics like multiplicative genera and cobordism that aren't handled in the default algebraic geometry class.
  • $\begingroup$ May’s book does not have more characteristic classes than Milnor’s book. $\endgroup$ – Zee Aug 14 '18 at 23:35

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