Introduction

Abstract

In the topological investigation of isolated critical points of complex-analytic functions the problem arises of describing the topology of its level sets. The topology of the level sets or infra-level sets of smooth real-valued functions on manifolds may be investigated with the help of Morse theory (see [255]). The idea there is to study the change of structure of infra-level sets and level sets of functions upon passing critical values. In the complex case passing through a critical value does not give rise to an interesting structure, since all the non-singular level sets near one critical point are not only homeomorphic but even diffeomorphic. The complex analogue of Morse theory, describing the topology of level sets of complex analytic functions, is the theory of Picard-Lefschetz (which historically precedes Morse theory). In Picard-Lefschetz theory the fundamental principle is not passing through a critical point but going round it in the complex plane.KeywordsRiemann SurfaceHomology GroupIntersection NumberMorse TheoryMonodromy GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.