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
Best Answer
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."