English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let M and N be topological spaces such that M admits a free involution $$\tau $$τ. A homotopy class $$\beta \in [ M , N ] $$β∈[M,N] is said to have the Borsuk–Ulam property with respect to$$\tau $$τ if for every representative map $$f:\,M \rightarrow N$$f:M→N of $$\beta $$β, there exists a point $$x \in M$$x∈M such that $$f ( \tau ( x) ) = f(x)$$f(τ(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 $$\tau $$τ. If $$M=N$$M=N is either the 2-torus $$\mathbb {T}^2$$T2 or the Klein bottle $$\mathbb {K}^2$$K2, we then solve the problem of deciding which homotopy classes of [M, M] have the Borsuk–Ulam property. First, if $$\tau :\,\mathbb {T}^2\rightarrow \mathbb {T}^2$$τ:T2→T2 is a free involution that preserves orientation, we show that no homotopy class of $$[ \mathbb {T}^2, \mathbb {T}^2]$$[T2,T2] has the Borsuk–Ulam property with respect to $$\tau $$τ. Second, we prove that up to a certain equivalence relation, there is only one class of free involutions $$\tau :\,\mathbb {T}^2\rightarrow \mathbb {T}^2$$τ:T2→T2 that reverse orientation, and for such involutions, we classify the homotopy classes in $$[\mathbb {T}^2, \mathbb {T}^2]$$[T2,T2] that have the Borsuk–Ulam property with respect to $$\tau $$τ in terms of the induced homomorphism on the fundamental group. Finally, we show that if $$\tau :\,\mathbb {K}^2\rightarrow \mathbb {K}^2$$τ:K2→K2 is a free involution, then a homotopy class of $$[\mathbb {K}^2, \mathbb {K}^2]$$[K2,K2] has the Borsuk–Ulam property with respect to $$\tau $$τ if and only if the given homotopy class lifts to the torus.

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