A Note on Morse Theory

Morse theory could be very well be called critical point theory. The idea is that by understanding the critical points of a smooth function on your manifold, you can recover the topology of your space. This basic idea has blossomed into many Morse theories. For instance, Robin Forman developed a combinatorial adaptation called discrete morse theory. We also have Morse-Bott theory, where we consider smooth functions on a manifold whose critical set is a closed submanifold. As a final example, Edward Witten used deformation of a differential and harmonic forms to produce a Morse homology. In this note, we will provide the basic definitions and theorems of Morse theory. We’ll conclude with a discussion of Morse homology. The stress will be on examples and understanding the ideas, rather than approaching the details of proofs.