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 $BS(M)$ ?

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