2013/04/29 by Michael Friedman, Friedman, Michael, David Garber +1
Computer Science · Mathematics · #14H30 #Algebraic Topology (math.AT) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematics and Applications #math.AT #math.GR #math.GT #msc:14H30
paper · pdf · doi:10.48550/arxiv.1304.7561
25 pages, 19 figures; Originally was a part of the paper arXiv:1111.5291, but is now separated; submitted
arxiv created 2013/04/29 · openalex publication_date 2013/04/29 · arxiv updated 2013/04/30 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
The fundamental group of the complement of a plane curve is a very important topological invariant. In particular, it is interesting to find out whether this group is determined by the combinatorics of the curve or not, and whether it is a direct sum of free groups and a free abelian group, or it has a conjugation-free geometric presentation. In this paper, we investigate the structure of this fundamental group when the graph of the conic-line arrangement is a unique cycle of length n and the conic passes through all the multiple points of the cycle. We show that if n is odd, then the affine fundamental group is abelian but not conjugation-free. For the even case, if n>4, then using quotients of the lower central series, we show that the fundamental group is not even a direct sum of a free abelian group and free groups.