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Algebraic and topological properties of the braid groups of the real projective plane, sphere, disk, orbit configuration spaces, and relations with crystallographic groups

Grant number: 14/50131-7
Support Opportunities:Regular Research Grants
Duration: April 01, 2014 - March 31, 2016
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Convênio/Acordo: CNRS
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 concerned with algebraic & topological properties of configuration spaces & braid groups, notably those of the disc D^2, the sphere S^2& the projective plane RP^2, some generalizations, such as orbit configuration spaces, & the relations of these braid groups with the classical crystallographic groups. The braid groups & their generalizations have attracted much attention, partly due to the fact that these groups arise in various contexts (topology, knot theory, algebra) Our proposal focusses on 3 different aspects: the homotopy fibres of inclusions of the configuration spaces of S ^2 & RP^2 in the Cartesian product, and orbit configuration spaces; the conjugacy classes of the finite subgroups of the braid groups of RP^2; & the relations between the braid groups of D^2, S^2, RP^2 and crystallographic groups. We will study these problems using algebra, algebraic topology, topology geometry. The results should have consequences for the associated mapping class groups. (AU)

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Scientific publications (6)
(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)
GONCALVES, DACIBERG LIMA; GUASCHI, JOHN; LAASS, VINICIUS CASTELUBER. The Borsuk-Ulam property for homotopy classes of self-maps of surfaces of Euler characteristic zero. Journal of Fixed Point Theory and Applications, v. 21, n. 2, . (14/50131-7)
GOLASINSKI, MAREK; GONCALVES, DACIBERG LIMA; GUASCHI, JOHN. On the homotopy fibre of the inclusion map F-n (X) hooked right arrow Pi(n)(1) X for some orbit spaces X. BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA, v. 23, n. 1, p. 29-pg., . (14/50131-7, 12/24454-8)
GONCALVES, DACIBERG LIMA; GUASCHI, JOHN; OCAMPO, OSCAR. Almost-crystallographic groups as quotients of Artin braid groups. Journal of Algebra, v. 524, p. 160-186, . (14/50131-7, 16/24707-4, 16/50354-1, 12/24454-8)
GONCALVES, DACIBERG LIMA; GUASCHI, JOHN. A Survey of the Homotopy Properties of Inclusion of Certain Types of Configuration Spaces into the Cartesian Product. CHINESE ANNALS OF MATHEMATICS SERIES B, v. 38, n. 6, p. 1223-1246, . (14/50131-7, 12/24454-8)
GONCALVES, DACIBERG LIMA; GUASCHI, JOHN. INCLUSION OF CONFIGURATION SPACES IN CARTESIAN PRODUCTS, AND THE VIRTUAL COHOMOLOGICAL DIMENSION OF THE BRAID GROUPS OF S-2 AND RP2. PACIFIC JOURNAL OF MATHEMATICS, v. 287, n. 1, p. 71-99, . (14/50131-7)
GOLASINSKI, MAREK; GONCALVES, DACIBERG LIMA; GUASCHI, JOHN. On the homotopy fibre of the inclusion map F-n (X) hooked right arrow Pi(n)(1) X for some orbit spaces X. BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA, v. 23, n. 1, SI, p. 457-485, . (14/50131-7, 12/24454-8)

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