### 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.

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.

#### Keywords

Critical points/values; degree of maps; algebraic $\varphi$-category

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