# Betti sum, cup-length, Lusternik-Schnirelmann category, and critical points

I am trying to make some order in the notions of cup-length, sum of Betti numbers, LS category, critical points of functions.

Let $$M$$ be smooth closed compact manifold. We denote by Crit($$M$$) the minimal number of critical points of a smooth function on $$M$$, $$BS(M)$$ the sum of Betti numbers, CL(M) its cup-length, and $$LS(M)$$ its LS category.

What is known about the relations between these notions ?

It seems that $$BS(M) \geq CL(M)$$, and it was proved that $$Crit(M) \geq LS(M) \geq CL(M)$$. Am I mistaken here ?

Is something kown about the relation between $$LS(M)$$ and $$BS(M)$$ ?

• At the very least, you have $BS(M) \geq Crit(M)$ when $M$ is a simply connected manifold of dimension at least 5, by handle cancellation techniques a la Smale's proof of the h-cobordism theorem.
– mme
Oct 17, 2018 at 14:52
• The comment describes how to construct a Morse function on $M$ with $BS(M)$ critical points.
– mme
Oct 17, 2018 at 15:03
• Combining that with the inequality $Crit(M) \geq LS(M)$ which is already in your post gives $BS(M) \geq LS(M)$. Observe that this inequality is not true for arbitrary manifolds: On page 354 of this article, it is explained that $LS(P) = 3$, where $P$ is the Poincare homology sphere, which has $BS(P) = 2$.
– mme
Oct 17, 2018 at 15:19
• Yes, many. See the book "Lusternik-Schnirelmann category" by Cornea-Lupton-Oprea-Tanre for a good discussion of this. It is a good reference for many related topics. In particular, Proposition 7.26 states that a closed connected manifold of dimension $n$ has $Crit(M) \leq n + 1$. So, for instance, every surface of genus $g$ admits a smooth function with three critical points.
– mme
Oct 17, 2018 at 15:51