Free involutions on torus semi-bundles and the Borsuk-Ulam Theorem for maps into $\mathbf{R}^n$

The Borsuk-Ulam theorem for closed 3-dimensional manifolds having Nil Geometry

“…This will allow us to compute the list of all pairs (M, τ ), where M belongs to this family and τ is a free involution on M . The procedure (same as in [2] and [3]) is the following: starting with a manifold N of the table above, we shall determine all double coverings Ñ → N by a systematic use of Reidemeister-Schreier algorithm ( [5]), and find that Ñ again belongs to the family. We obtain a list of double coverings M → N , indexed by N .…”

The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle - part 2

“…One such generalisation consists in the study of the validity of the theorem when we replace S n and R n by manifolds M and N respectively, and we replace the antipodal map of S n by a free involution τ of M. More precisely, the triple (M, τ ; N) is said to have the Borsuk-Ulam property if for any continuous map f : M −→ N , there exists a point x ∈ M for which f (τ (x)) = f (x). Some examples of results in this direction may be found in [1,5,9,10,13]. Very recently, the following more refined Borsuk-Ulam-type problem was introduced by the authors in the context of homotopy classes of maps from M to N [11].…”