Topological classification of generic real meromorphic functions

TL;DR

This paper introduces a combinatorial tool called the 'park' to classify the topology of generic real meromorphic functions, establishing a one-to-one correspondence with connected components and calculating associated Hurwitz numbers.

Contribution

It defines the 'park' as a new combinatorial invariant that completely determines the topological type of generic real meromorphic functions and links it to the structure of their moduli space.

Findings

The 'park' uniquely classifies the topological types of functions.

The set of all parks corresponds bijectively to connected components in the function space.

Hurwitz numbers are computed for each component.

Abstract

In this article, to each generic real meromorphic function (i.e., having only simple branch points in the appropriate sense) we associate a certain combinatorial gadget which we call the park of a function. We show that the park determines the topological type of the generic real meromorphic function and that the set of all parks is in $1-1$-correspondence with the set of all connected components in the space of generic real meromorphic functions. For any of the above components, we introduce and calculate the corresponding Hurwitz number. Finally as a consequence of our results we determine the topological types of real meromorphic functions from monodromy of orbifold coverings.