The size of some critical sets by means of dimension and algebraic $\varphi$-category

Abstract

Let $M^n$, $N^n$, $n\geq 2$, 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 $f\colon M\to N$, under the assumption on the index $[\pi_1(N):{\rm Im}(f_*)]$ to be infinite. By contrast with the described situation one shows, after some estimates on the algebraic $\varphi$-category of some pairs of finite groups, that a critical set of smaller dimension keeps the degree away from zero.