The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero
Résumé
Let M and N be topological spaces such that M admits a free involution τ. A homotopy class β ∈ [M, N ] is said to have the Borsuk-Ulam property with respect to τ if for every representative map f : M → N of β, there exists a point x ∈ M such that 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 τ. If M = N is either the 2-torus T^2 or the Klein bottle K^2 , we then solve the problem of deciding which homotopy classes of [M, M ] have the Borsuk-Ulam property. First, if τ : T^2 → T^2 is a free involution that preserves orientation, we show that no homotopy class of [T^2 , T^2 ] has the Borsuk-Ulam property with respect to τ. Secondly, we prove that up to a certain equivalence relation, there is only one class of free involutions τ : T^2 → T^2 that reverse orientation, and for such involutions, we classify the homotopy classes in [T^2 , T^2 ] that have the Borsuk-Ulam property with respect to τ in terms of the induced homomorphism on the fundamental group. Finally, we show that if τ : K^2 → K^2 is a free involution, then a homotopy class of [K^2 , K^2 ] has the Borsuk-Ulam property with respect to τ if and only if the given homotopy class lifts to the torus.
Domaines
Topologie géométrique [math.GT]
Origine : Fichiers produits par l'(les) auteur(s)
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