Generalisations of configuration spaces, relations between braid and almost-crystallographic groups, and applications to the study of the Borsuk-Ulam property and multi-valued maps

• Grant number: 16/50354-1
• Support Opportunities: Regular Research Grants
• Start date: February 01, 2017
• End date: January 31, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Agreement: CNRS
• 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: 12/24454-8 - Algebraic, geometric and differential topology, AP.TEM

Abstract

Our project is related to various aspects of the theory of braid groups, configuration spaces and their generalisations. In problem (1), we shall analyse the homotopy type and the fundamental group of certain orbit configuration spaces, notably for free $Z_2$ actions, and of graph configuration spaces. In problem (2), we investigate the connections between braid and almost-crystallographic groups, as well as generalizations involving lower central series and surface braid groups. In the remaining part of the project, we study applications to two problems in which surface braid groups arise naturally. In problem (3), we exploit a formulation of the Borsuk-Ulam property in terms of braid equations, notably for maps between surfaces, the aim being to decide whether given homotopy classes of maps have this property or not. The second application, in problem (4), is to fixed point and coincidence theory of $n$-valued maps, in particular for maps between surfaces. We intend to explore conditions for $n$-valued maps to be deformable to fixed point free (or coincidence free) maps, as well as whether they satisfy the Wecken property. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

BELLINGERI, PAOLO; GONCALVES, DACIBERG LIMA; GUASCHI, JOHN. ower central series, surface braid groups, surjections and permutation. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, v. 172, n. 2, p. 373-399, MAR 2022. (16/50354-1)

GONCALVES, DACIBERG LIMA; GUASCHI, JOHN; OCAMPO, OSCAR. Almost-crystallographic groups as quotients of Artin braid groups. Journal of Algebra, v. 524, p. 160-186, APR 15 2019. (14/50131-7, 16/24707-4, 16/50354-1, 12/24454-8)