The size of some critical sets by means of dimension and algebraic \(\varphi\)-category (Q1958999)

The author investigates the size of some critical sets of a \(C^1\) function \(f:M\rightarrow N\), where \(M^n\) and \(N^n\) with \(n\geq 2\) are two compact connected manifolds. Roughly speaking, among the others results, he proves that if \(f\) is a mapping acting between manifolds with infinite algebraic \(\varphi-\)category of their fundamental groups, then \(f\) has zero degree and, consequently, high dimensional critical sets.