Circle-valued Morse theory

Circle-valued Morse theory is an extension of classical Morse theory, which is a mathematical framework used primarily in differential topology and critical point theory to study the topology of manifolds via smooth functions. In essence, classical Morse theory analyzes the critical points of real-valued functions on manifolds to glean information about the manifold's topology. In circle-valued Morse theory, instead of considering real-valued functions, one studies functions that map a manifold into the circle \( S^1 \).