On the Fundamental Group of Plane Curve Complements

Author

Mitchell Scofield, Boise State UniversityFollow

Publication Date

5-2019

Date of Final Oral Examination (Defense)

3-1-2019

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Major Advisor

Jens Harlander, Ph.D.

Advisor

Zachariah Teitler, Ph.D.

Advisor

Uwe Kaiser, Ph.D.

Abstract

Given a polynomial f(x,y) monic in y of degree d, we study the complement ℂ²-C, where C is the curve defined by the equation f(x,y)=0. The Zariski-Van Kampen theorem gives a presentation of the fundamental group of the complement ℂ²-C. Let NT be be the set of complex numbers x for which f(x,y) has multiple roots (as a polynomial in y). Let : ℂ − NT → ℂ^d − Δ be the map that sends x to the d-tuple of distinct roots (Δ is the diagonal in ℂ^d). It induces a map ∇: F_r → B_d on the fundamental group level, where F_r is the free group on r letters and B_d is the braid group on d strands. In order to write down the Zariski-Van Kampen presentation one needs an explicit understanding of ∇. This is hard to come by in general. It turns out that under special circumstances ∇ can be computed directly from combinatorial and visual (real) information on the curve C. The method in these special situations is similar to the computation of the presentation of the fundamental group of a knot complement in ℝ³.

DOI

10.18122/td/1538/boisestate

Recommended Citation

Scofield, Mitchell, "On the Fundamental Group of Plane Curve Complements" (2019). Boise State University Theses and Dissertations. 1538.
10.18122/td/1538/boisestate