Parameterized versions of the Borsuk-Ulam theorem

• Grant number: 12/03316-6
• Support Opportunities: Regular Research Grants
• Start date: June 01, 2012
• End date: May 31, 2014
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Thaís Fernanda Mendes Monis
• Grantee: Thaís Fernanda Mendes Monis
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil

Abstract

The classical Borsuk-Ulam theorem states that for every continuous map f defined over the m-dimensional sphere, S m, and with values in the m-dimensional Euclidean space, R m, the set B(f) of the points x such that f(x)=f(-x) is non-empty. The set B(f) will called the set of the Borsuk-Ulam solutions to f. In a paper published recently, Profesor Stanislaw Spiez together T. Schick, R. S. Simon and H. Torunczyk proved that for a "continuous" family of maps f_w : S m \to’ R m, parameterized by points of a compact manifold W (with or without boundary), its Borsuk-Ulam solution set also depends "continuously" on the parameter space W. "Continuity" here means that the solution set supports a homology class which maps onto the fundamental class of W. To prove the above result, a relative squaring construction in Cech homology is used. Itseems that this construction can be useful in other contexts and therefore deserves independentinterest. However, the proof of this result heavily depend on the assumption that W is amanifold and the methods used there do not work in a more general case. We plan to generalize the above theorem to a more general setting by using another approach. In particular, we believe that it should be possible to relax the assumption that W is a manifoldto a more general class of compacta. (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)

MONIS, THAIS F. M.; SPIEZ, STANISLAW. Lefschetz coincidence class for several maps. Journal of Fixed Point Theory and Applications, v. 18, n. 1, p. 61-76, MAR 2016. (12/03316-6, 13/07936-1)