Morse

Initially introduced by Marston Morse in the early 20th century, this theory focuses on understanding the topology of a manifold by studying smooth functions defined on it and their critical points. The central idea is that the critical points of a smooth function on a manifold reveal important topological features of the manifold, and the flow lines of the gradient of that function provide a way to move between critical points.

Witten

In the 1980s, physicist Edward Witten introduced a reinterpretation of Morse theory within the framework of supersymmetric quantum mechanics and quantum field theory. This reformulation connects Morse theory to quantum mechanics, re-casting Morse's results in terms of the existence of a chain complex, built from the critical points of a function and the gradient flow lines connecting them, and whose homology computes ordinary homology.

Floer

Introduced by Andreas Floer in the 1980s, Floer theory further refines the Morse-theoretic approach by providing more detailed information about the manifold. This involves using gradient flow trajectories than are encoded in ordinary homology. Additionally, Floer suggested that the proper framework for studying morse theory is stable homotopy theory