Abstract
Let M and N be topological spaces such that M admits a free involution \(\tau \). A homotopy class \(\beta \in [ M , N ] \) is said to have the Borsuk–Ulam property with respect to \(\tau \) if for every representative map \(f:\,M \rightarrow N\) of \(\beta \), there exists a point \(x \in M\) such that \(f ( \tau ( x) ) = f(x)\). 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 \(\tau \). If \(M=N\) is either the 2-torus \(\mathbb {T}^2\) or the Klein bottle \(\mathbb {K}^2\), we then solve the problem of deciding which homotopy classes of [M, M] have the Borsuk–Ulam property. First, if \(\tau :\,\mathbb {T}^2\rightarrow \mathbb {T}^2\) is a free involution that preserves orientation, we show that no homotopy class of \([ \mathbb {T}^2, \mathbb {T}^2]\) has the Borsuk–Ulam property with respect to \(\tau \). Second, we prove that up to a certain equivalence relation, there is only one class of free involutions \(\tau :\,\mathbb {T}^2\rightarrow \mathbb {T}^2\) that reverse orientation, and for such involutions, we classify the homotopy classes in \([\mathbb {T}^2, \mathbb {T}^2]\) that have the Borsuk–Ulam property with respect to \(\tau \) in terms of the induced homomorphism on the fundamental group. Finally, we show that if \(\tau :\,\mathbb {K}^2\rightarrow \mathbb {K}^2\) is a free involution, then a homotopy class of \([\mathbb {K}^2, \mathbb {K}^2]\) has the Borsuk–Ulam property with respect to \(\tau \) if and only if the given homotopy class lifts to the torus.
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Acknowledgements
Part of this work is contained in the Ph.D. thesis [11] of the third author who was supported by CNPq Project no 140836 and Capes/COFECUB Project no. 12693/13-8. The first and second authors were partially supported by CNRS/FAPESP Project numbers 2014/50131-7 (Brazil) and 2014/226555 (France). The second author wishes to thank the ‘Réseau Franco-Brésilien en Mathématiques’ for financial support during his visit to the Instituto de Matemática e Estatística, Universidade de São Paulo, from the 9th of July to the 1st of August 2016.
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Gonçalves, D.L., Guaschi, J. & Laass, V.C. The Borsuk–Ulam property for homotopy classes of self-maps of surfaces of Euler characteristic zero. J. Fixed Point Theory Appl. 21, 65 (2019). https://doi.org/10.1007/s11784-019-0693-z
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DOI: https://doi.org/10.1007/s11784-019-0693-z