Morse Theory. It's a beautiful construction of a cell complex from a Morse function on a manifold. As a result, there are inequalities estimating the number of critical points by ranks of homology groups.
I'm new to this theory, so I would like to build up my basic on it. Also does anyone know where can I find relevant information on the topic above?
"Morse Theory", by John Milnor. Perhaps one of the best reads in all of mathematics.