In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses.

They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead Elements of Homotopy Theory. These are both excellent books that (theoretically) give you overviews and introduction to most of the main topics that you need for becoming a modern researcher in algebraic topology.

Differential Geometry seems replete with excellent introductory textbooks. From Lee to do Carmo to so many others.

Now you might be thinking that Kobayashi/Nomizu seems natural. But the age of those books is showing in terms of what people are really doing today compared to what you learn from using those books. They just aren't the most efficient way to learn modern differential geometry (or so I've heard).

I am looking for a book that covers topics like Characteristic Classes, Index Theory, the analytic side of manifold theory, Lie groups, Hodge theory, Kahler manifolds and complex geometry, symplectic and Poisson geometry, Riemmanian Geometry and geometric analysis, and perhaps some relations to algebraic geometry and mathematical physics. But none of these topics completely, just as Switzer does with a unifying perspective and proofs of legitimate results done at an advanced level, but really as an introduction to each of the topics (Switzer does this with K-theory, spectral sequences, cohomology operations, Spectra...).

The only book I have found that is sort of along these lines is Nicolaescu's Lectures on the Geometry of Manifolds, but this book misses many topics.

This was inspired by page viii of Lee's excellent book: link where he lists some of these other topics and almost implies that they would take another volume. I'm wondering whether that advanced volume exists.

Any recommendations for great textbooks/monographs would be much appreciated!


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Of course this is not the book that you are looking for, since it covers only one topic, but thoroughly and it is a classic : MILNOR -STASHEFF "Characteristic Classes"

A nice and complete book on Complex Geometry is that of WELLS - GARCIA PRADA. "Differential Analysis on Complex Manifolds", where you may find Complex Characteristic Classes (Chern Classes), and Hodge theory, besides Elliptic Operators.


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