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Grassmannians are a pretty useful subject in numerous fields of mathematics (and physics). In fact, it was the first non-trivial higher-dimensional example that was given in an introductory projective geometry course during my education.

Later I learned you can use them to define universal bundles and that they are playing a role in higher-dimensional geometry and topology. Though I have never came across a book or a survey article on the geometry and topology of those beasts. The field is a little wide, so let me specify what I am interested in:

  • Topology and Geometry of Grassmannians $G_k(\mathbb{R}^n)$ or $G_k(\mathbb{C}^n)$.
  • Connections with bundle and obstruction theory.
  • Differential Topology of $G_k(\mathbb{R}^n)$ or $G_k(\mathbb{C}^n)$ (for instance, are there exotic Grassmannians).
  • Homotopy Theory of $G_k(\mathbb{R}^n)$ or $G_k(\mathbb{C}^n)$.
  • Algebraic Geometry of $G_k(V)$, where $V$ is a $n$-dimensional vectorspace over a (possible characteristic $\ne 0$ field $\mathbb{F}$)

So, are there books or survey articles on those subjects.

Edit: I just found a thesis on the subject. Some of these questions are addressed here: http://www.math.mcgill.ca/goren/Students/KolhatkarThesis.pdf

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2 Answers 2

up vote 8 down vote accepted

I enjoyed the book by Milnor and Stasheff, "Characteristic Classes." This explains the business of the universal bundle, and the cohomology ring (which is to say, characteristic classes).

As for the algebraic case...this is explained in the new edition of EGA I, but it is a little technical (relying on a standard lemma in basic moduli theory, which I found rather difficult to understand). There are also explanations in the book "FGA Explained."

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+1 - Milnor & Stasheff definitely! –  Juan S Aug 25 '11 at 0:34
    
Thank you, I did actually enjoyed Milnor and Stasheff. Are there any differential topological classification results of Grassmannians, as you know of ? –  Willem Noorduin Aug 26 '11 at 6:47
    
The Milnor Stasheff is sadly incomplete. A better source would be more helpful at here. –  Bombyx mori Sep 16 '12 at 3:41

This is an answer to your continued question on MO. http://mathoverflow.net/questions/73736/topology-and-geometry-of-grassmannians-g-k-mathbbrn-or-g-k-mathbbcn

(1) What would you want out of a "topological classification"?

(2) Yes, Grassmannians can have exotic smooth structures. For example $\mathbb RP^n = G_1(\mathbb R^{n+1})$ has well-known exotic smooth structures for various $n$.

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I want to know from a (differential) topological classification what you can say what Grassmannians do fall in the known classification (for example orientable/nonorientable, elliptic fibrated or not, K3 or not, etcetera), with the structure as a Grassmannian in the back of your head. –  Willem Noorduin Aug 26 '11 at 22:09
    
To me an exotic copy of $M^n$ is a manifold $X^n$ which is homeomorphic, but not diffeomorphic to $M^n$. A fake copy of $M^n$ is a manifold $Y^n$ which is homotopic, but not homeomorphic to $M^n$ (fakeness is weaker than exoticness, in a way), but I realize the terminology differs from topologist to topologist. –  Willem Noorduin Aug 26 '11 at 22:17
    
@Willem: Right, sorry I forgot that terminology. But projective spaces do carry exotic smooth structures. For example, if $n$ is odd and there's an element of order $4$ in the group of homotopy $n$-spheres. –  Ryan Budney Aug 26 '11 at 22:46
    
@Willem: what do you mean by "known classification"? It would help if you make that precise. –  Ryan Budney Aug 26 '11 at 22:46
    
Thank you for your answer. By known classification I mean (in case $n = 4$) for example the classification of complex compact surfaces (I know this has still open ends). Of course for $n > 4$ this is a bit wide, I guess, but then you can ask (for example) what van be said about $\pi(G_k(\mathbb{R}^n)$), where $\pi$ is for instance the Plucker embedding. –  Willem Noorduin Aug 26 '11 at 23:35

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