Monodromy Groups of Critical Points

Abstract

Morse theory studies the restructurings, perestroikas, or metamorphoses that the level set f^-1(x) of a real function f: M → ℝ, defined on a manifold M, undergoes as x passes through the critical values of f. The Picard-Lefschetz theory is the complex analogue of Morse theory. In the complex case the set of critical values does not divide the range ℂ of a complex-valued function into connected components, and no restructurings occur: all level manifolds close to a critical one are topologically identical. For this reason, in the complex case, rather than passing through a critical value, one has to go around it in the plane ℂ where the function takes its values.