## Publication Date

8-2023

## Date of Final Oral Examination (Defense)

6-23-2023

## Type of Culminating Activity

Thesis

## Degree Title

Master of Science in Mathematics

## Department

Mathematics

## Supervisory Committee Chair

Jens Harlander, Ph.D.

## Supervisory Committee Member

Uwe Kaiser, Ph.D.

## Supervisory Committee Member

John Clemens, Ph.D.

## Abstract

From a *d*-sheeted branched covering *f* : *M* β *N*, where *M* and *N* are surfaces, one can read off the branch datum

*D*(*f*) = {*M*, *N*, *d*, (*A*_{1}, . . . , *A _{r}*)},

where *A _{i}* = [

*e*1, . . . ,

_{i}*e*] is a partition of

_{in_i}*d*. Furthermore, a relationship between the Euler characteristics between

*M*and

*N*is known, called the Riemann-Hurwitz formula

π³(*M*) = *d*π³(*N*) β *v*(*D*(*f*)),

where *v*(*D*(*f*)) = β(*e _{ij}* β 1). The Hurwitz problem asks for a characterization of realizable abstract branching data. More precisely, given an abstract branch datum

*D* = {*M*, *N*, *d*, (*A*_{1}, . . . , *A _{r}*)},

where

π³(*M*) = *d*π³(*N*) β *v*(*D*),

how do we know that there is a branched covering *f* : *M* β *N* such that *D* = *D*(*f*)?

Hurwitz himself found an answer (at least for *N* = *S*^{2}) in terms of permutations Ξ±_{1}, . . . , Ξ±_{r} β *S _{d}*, whose cycle structures realize the given partitions: the cycle structure of Ξ±

_{i}is

*A*.

_{i}Eighty years later Gersten solved the Hurwitz problem, at least in the case *M* = *N* = *S*^{2}, using cancellation diagrams (see fig.1). He showed that the datum *D* = {*S*^{2}, *S*^{2}, *d*, (*A*_{1}, . . . , *A _{r}*)}, where π³(

*S*

^{2}) =

*d*π³(

*S*

^{2}) β

*v*(

*D*), that is 2 = 2

*d*β

*v*(

*D*), is realizable if a cancellation diagram exists for a family of words in the group

Ξ = < *x*_{1}, . . . , *x _{r} | x_{1} . . . x_{r} *= 1 >

that is determined by the given partitions. The appeal of Gersten's solution lies in the fact that the cancellation diagram depicts the actual branched covering. Thus, his solution is more concrete than the solution of Hurwitz.

In this thesis we discuss and compare both solution methods, showcasing everything through examples. We present an example of a realizable branch datum which is used to illustrate methods throughout the paper.

## DOI

https://doi.org/10.18122/td.2096.boisestate

## Recommended Citation

Byars, James Alexander, "A History of the Hurwitz Problem Concerning Branched Coverings" (2023). *Boise State University Theses and Dissertations*. 2096.

https://doi.org/10.18122/td.2096.boisestate