The Borsuk-Ulam theorem for general spaces (Q1423766)

Let \(X\) and \(Y\) be two topological spaces, and let \(T: X\to X\) and \(S: Y\to Y\) be free involutions on \(X\) and \(Y\), respectively. A map \(f\colon X\to Y\) is said to be equivariant if \(Sf= fT\). The author shows some conditions on the homology groups of \(X\) and \(Y\) to ensure that there is no equivariant map between them. Moreover, by using such results, the authors find a condition under which every map \(f: X\to Y\) has a \(T\)-coincidence point, i.e., a point \(x\in X\) with \(f(x)=f(T(x))\). Of course, the involution \(S\) is omitted in this case.