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, (A1, . . . , Ar)},
where Ai = [ei1, . . . , ein_i] is a partition of d. Furthermore, a relationship between the Euler characteristics between M and N is known, called the Riemann-Hurwitz formula
π³(M) = dπ³(N) β v(D(f)),
where v(D(f)) = β(eij 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, (A1, . . . , Ar)},
where
π³(M) = dπ³(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 = S2) in terms of permutations Ξ±1, . . . , Ξ±r β Sd, whose cycle structures realize the given partitions: the cycle structure of Ξ±i is Ai.
Eighty years later Gersten solved the Hurwitz problem, at least in the case M = N = S2, using cancellation diagrams (see fig.1). He showed that the datum D = {S2, S2, d, (A1, . . . , Ar)}, where π³(S2) = dπ³(S2) β v(D), that is 2 = 2d β v(D), is realizable if a cancellation diagram exists for a family of words in the group
Ξ = < x1, . . . , xr | x1 . . . xr = 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