Final Examination
1. A cell structure for the Grassmannians G_k(R^n)
and G_k(C^n) of k-planes through the
origin in real (complex) n-space. In particular,
for K=R or C, explore
the duality between G_k(K^(n+k)) and
G_n(K^(n+k)), and what happens to the cell
structure of G_k(K^n) as
n-->infinity?
[See Milnor and Stasheff,
Characteristic Classes, Chapter 6, and also Griffiths and
Harris, Principles of Algebraic Geometry].
2. How can you show that S^2XS^2 is not the same as
the connected sum of two copies of CP^2? They are both
4-dimensional manifolds, and have the same homology groups.
This is a ``take home'' examination. Please turn in your
papers by May 6, 1997. These questions are open-ended; you
may consult any published works, but please give clear
references. If you must ask other people for help, please
let me know. The format of your ``answers'' should be expository,
as if you were planning a lecture on the topic to your fellow
students. Please write clearly.