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.
Citation
Cornel Pintea. "The size of some critical sets by means of dimension and algebraic $\varphi$-category." Topol. Methods Nonlinear Anal. 35 (2) 395 - 407, 2010.
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