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(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

The Borsuk-Ulam property for homotopy classes of self-maps of surfaces of Euler characteristic zero

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Author(s):
Goncalves, Daciberg Lima [1] ; Guaschi, John [2] ; Laass, Vinicius Casteluber [3]
Total Authors: 3
Affiliation:
[1] Univ Sao Paulo, IME, Dept Matemat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil
[2] Normandie Univ, UNICAEN, CNRS, Lab Math Nicolas Oresme UMR CNRS 6139, F-14000 Caen - France
[3] Univ Fed Bahia, Dept Matemat, IME, Av Adhemar Barros S-N, BR-40170110 Salvador, BA - Brazil
Total Affiliations: 3
Document type: Journal article
Source: Journal of Fixed Point Theory and Applications; v. 21, n. 2 JUN 2019.
Web of Science Citations: 0
Abstract

Let M and N be topological spaces such that M admits a free involution . A homotopy class {[}M,N] is said to have the Borsuk-Ulam property with respect to if for every representative map f:MN of , there exists a point xM such that f((x))=f(x). In the case where M is a compact, connected manifold without boundary and N is a compact, connected surface without boundary different from the 2-sphere and the real projective plane, we formulate this property in terms of the pure and full 2-string braid groups of N, and of the fundamental groups of M and the orbit space of M with respect to the action of . If M=N is either the 2-torus T2 or the Klein bottle K2, we then solve the problem of deciding which homotopy classes of {[}M,M] have the Borsuk-Ulam property. First, if :T2T2 is a free involution that preserves orientation, we show that no homotopy class of {[}T2,T2] has the Borsuk-Ulam property with respect to . Second, we prove that up to a certain equivalence relation, there is only one class of free involutions :T2T2 that reverse orientation, and for such involutions, we classify the homotopy classes in {[}T2,T2] that have the Borsuk-Ulam property with respect to in terms of the induced homomorphism on the fundamental group. Finally, we show that if :K2K2 is a free involution, then a homotopy class of {[}K2,K2] has the Borsuk-Ulam property with respect to if and only if the given homotopy class lifts to the torus. (AU)

FAPESP's process: 14/50131-7 - Algebraic and topological properties of the braid groups of the real projective plane, sphere, disk, orbit configuration spaces, and relations with crystallographic groups
Grantee:Daciberg Lima Gonçalves
Support Opportunities: Regular Research Grants