A GENERALIZATION OF THE BORSUK-ULAM THEOREM ON MANIFOLDS

Let M be a closed n-dimensional manifold. We prove for any continuous map f : M ! Rn and odd degree map g : M ! Sn there are points x; y 2 M such that f(x) = f(y) and g(x) = ¡g(y). Clearly it implies Borsuk-Ulam theorem when M = Sn and g be an identity map. In particular we give an easy and short proof, method based on cohomological properties of these spaces