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^2*X*S^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.