THE SIZE OF SOME CRITICAL SETS BY MEANS OF DIMENSION AND ALGEBRAIC φ-CATEGORY

Let M-n, N-n, n >= 2, be compact connected manifolds. We first observe that mappings of zero degree have high dimensional critical sets and show that the only possible degree is zero for maps f: M -> N, under the assumption on the index [pi(1)(N) : Im(f(*))] to be infinite. By contrast with the described situation one shows, after some estimates on the algebraic phi-category of some pairs of finite groups, that a critical set of smaller dimension keeps the degree away from zero.