Morse Theory and critical groups

Question

Please I have a question:

What is the relation between Morse theory and critical point theory ?

I studied the Morse inequalities and critical groups, but i can not not find or at least i do not understand the results of Morse given by critical groups and Morse inequalities .

Please

Thank you .

Answer 1

This Wikipedia article on Morse Theory describes what you are talking about fairly well.

Critical points are essential to Morse theory, because the topology of level sets changes at every critical point. The way it changes is encoded by the index of the critical point. Using all of the indices, one can compute an estimate on Betti numbers.

This may bit have answered your question; if it has not, please post more details, and I or someone else will help more.