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

Cornel Pintea


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.


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

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