Braid groups of the projective plane and Configuration orbit spaces

• Grant number: 24/19199-6
• Support Opportunities: Regular Research Grants
• Start date: February 01, 2025
• End date: January 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Mobility Program: SPRINT - Projetos de pesquisa - Mobilidade
• Principal Investigator: Daciberg Lima Gonçalves
• Grantee: Daciberg Lima Gonçalves
• Principal researcher abroad: John Guaschi
• Institution abroad: Université de Caen Basse-Normandie, France
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 22/16455-6 - Algebraic, geometric, and differential topology, AP.TEM

Abstract

The participants of the project are the two Principal Investigators, Daciberg LIMA GONÇALVES (Universidade de São Paulo) and John GUASCHI (Université de Caen Normandie). The project lies at the interface between topology, algebraic topology and algebra, and the main general areas are surface braid groups, configuration spaces and their generalisations. The scientific project consists in the exploration of configuration spaces of surfaces in two contexts. In the first, we consider the usual configuration spaces of the projective plane RP^2 whose fundamental group is the braid group B_n(RP^2). We propose to study the family of finite subgroups of B_n(RP^2), focussing on questions related to the structure of this family up to conjugation. In the second case, given a discrete group G acting freely on a surface S, we study the corresponding orbit configuration space F_n^G(S), which is a generalisation of the usual configuration spaces. One of the aims of the project is to determine the fundamental group for the cases where S is either the 2-sphere with a finite number of orbits removed for the action induced by the antipodal map, or is the Euclidean plane R^2 for which we consider the action by translations of Z+Z or via the Klein bottle group. (AU)