The Borsuk-Ulam theorem for general spaces

Let X, Y be topological spaces and $T : X \rightarrow X$ a free involution. In this context, a question that naturally arises is whether or not all continuous maps $f : X \rightarrow Y$ have a T-coincidence point, that is, a point $x \in X$ with $f (x) = f (T (x))$. If additionally Y is equipped with a free involution $S : Y \rightarrow Y$ , another question is concerning the existence of equivariant maps $f : (X, T) \rightarrow (Y, S)$. In this paper we obtain results of this nature under cohomological (homological) conditions on the spaces X and Y.