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

The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle

“…The following proposition summarises some results of [5,Section 4] regarding the structure of P 2 (K 2 ) and the action by conjugation by σ on this group. Proposition 3.1.…”

“…Remark 1.2. The bijection of Proposition 1.1 may be obtained using standard arguments in homotopy theory that are described in detail in [11,Chapter V,Corollary 4.4], and more briefly in [5,Theorem 4]. In our specific case, the bijection is defined as follows: given a homotopy class β ∈ [T 2 , K 2 ], there exists a pointed map f : (T 2 , * ) → (K 2 , * ) that gives rise to a representative of β if we omit the basepoints.…”

“…In [5], the Borsuk-Ulam problem for homotopy classes of maps between compact surfaces without boundary was studied, and the sets [T 2 , T 2 ] and [K 2 , K 2 ] whose elements possess the Borsuk-Ulam property were characterised. By [4,Theorem 12], for any involution τ : T 2 → T 2 , the triple (T 2 , τ ; K 2 ) does not have the Borsuk-Ulam property.…”

“…Using this information, in this paper we classify the homotopy classes of maps from T 2 to K 2 that have the Borsuk-Ulam property for the orientation-preserving free involution τ 1 of T 2 . Our approach, which we now describe, is similar to that used in [5]. First, as in [5,Theorems 12 and 19], we identify π 1 (T 2 , * ) and π 1 (K 2 , * ) with the free Abelian group Z ⊕ Z and the (non-trivial) semi-direct product Z ⋊ Z respectively.…”

“…Our approach, which we now describe, is similar to that used in [5]. First, as in [5,Theorems 12 and 19], we identify π 1 (T 2 , * ) and π 1 (K 2 , * ) with the free Abelian group Z ⊕ Z and the (non-trivial) semi-direct product Z ⋊ Z respectively. These identifications will be helpful in formulating the results and in making explicit computations.…”