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