The Topology of Morse Functions

Abstract

The present chapter is the heart of Morse theory, which is based on two fundamental principles. The “weak” Morse principle states that as long as the real parameter t varies in an interval containing only regular values of a smooth function f: M → ℝ, then the topology of the sublevel set f ≤ t is independent of t. We can turn this on its head and state that a change in the topology of f ≤ t is an indicator of the presence of a critical point.