Morse functions and cohomology of homogeneous spaces

Morse functions are useful tool in revealing the geometric formation of its domain manifolds M. They define the handle decompositions of M from which the additive homologies H * (M) may be constructed. In these lectures two further questions were emphasized. (1) How to find a Morse function on a given manifold? (2) From Morse functions can one derive the multiplicative coho-mology rather than the additive homology? It is not our intention here to make detailed studies of these question. Instead, we will illustrate by examples solutions to them for some classical manifolds as homogeneous spaces. I am very grateful to Piotr Pragacz for the opportunity to speak of the wonder that I have experienced with Morse functions, and for his hospitality during my stay in Warsaw. Thanks are also due to Dr. Marek Szyjewski for taking the lecture notes from which the present article was initiated, and to Dr. M. Borodzik for many improvements on the earlier version of the note.