Critical points and high homotopy groups Is there any known or interesting relation between critical points (possibly degenerate, or maybe only nondegenerate) of a function on a manifold and generators/relations of high homotopy groups? I only know about the relation to representations of a fundamental group....
 A: Bott periodicity (which computes all the homotopy group of a Lie group was proved by Morse Theory (so, the study of critical points, admittedly on the various path spaces, not quite the manifold itself). See the nice write-up by Aaron Mazel-Gee
A: I am not sure how it is related to your question, but it is possible to construct a mapping $f\in C^1(\mathbb{S}^{n+1},\mathbb{S}^n)$ for all $n\geq 4$, that generates a nontrivial element in $\pi_{n+1}(\mathbb{S}^n)$, but such that all points of the mapping are critical, that is $\operatorname{rank}Df(x)<n$ for all $x\in\mathbb{S}^{n+1}$. This construction generalizes also to other homotopy groups  of spheres. It has been done in [1] and is related to results in [2] and [3].
[1] P. Goldstein, P. Hajlasz, P. Pankka, Topologically nontrivial counterexamples to Sard's theorem Int. Mat. Res. Not. IMRN. 2018 (published online). (arXiv:1804.07658)
[2] L. Guth,
Contraction of areas vs. topology of mappings. Geom. Funct. Anal. 23 (2013), 1804-1902.
[3] S. Wenger, R. Young,
Lipschitz homotopy groups of the Heisenberg groups. Geom. Funct. Anal. 24 (2014), 387-402.
