# 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 Morse function determines the homotopy type of a smooth manifold, so it must in principle specify the higher homotopy groups, but I don't know of any explicit description using Morse theory. – Arun Debray Oct 1 '18 at 21:43
• One more roundabout idea is that the number of critical points give bounds on the Betti numbers via the Morse inequalities, which in turn suggests information about the rational cohomology and therefore the Sullivan minimal model of the manifold, and ultimately the ranks of the rational homotopy groups. Of course all of this could be moot depending on the inequalities involved in any step of this... – Tobias Shin Oct 2 '18 at 1:42
• @TobiasShin Even knowing the Betti numbers of a closed manifold exactly does not help much in determining its rational homotopy groups, unless further information about the manifold is known (e.g. it is rationally elliptic), I think. Consider for example $S^2\times S^2 \times S^2$, which has only finitely many non-trivial rational homotopy groups, and $\mathbb{C}\mathbb{P}^3 \# \mathbb{C}\mathbb{P}^3 \# \mathbb{C}\mathbb{P}^3$ which has infinitely many non-trivial rational homotopy groups, yet the Betti numbers of these two manifolds coincide. – Aleksandar Milivojevic Oct 5 '18 at 22:51

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

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 $$\operatorname{rank}Df(x) for all $$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.