Spring 1997 MAT 539 Algebraic Topology

Final Examination

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.

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.