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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 $BS(M)$ ?

BrianT
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1 Answers1

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Just to clarify definitions and notation. Let $R$ be any (commutative) ring with unity, and $\mathbb{F}$ any field. Then:

$$\text{Crit}(M) := \min_{f \in C^\infty(M,\mathbb{R})} \left| \text{Crit}(f) \right|,$$

$$\text{LS}(M) := \min \left\{ k\geq 0 \mid M = U_0 \cup \dots \cup U_k, \, \text{where each } U_i \hookrightarrow M \text{ is nullhomotopic} \right\},$$

$$\text{CL}(M) := \max \left\{ k\geq 0 \mid \exists \, \alpha_1, \dots, \alpha_k \in H^{> 0}(M; R), \, \alpha_1 \smile \dots \smile \alpha_k \neq 0 \right\},$$

$$\text{BS}(M) := \sum_{i=0}^{\dim M} b_i(M), \quad \text{where } b_i(M) = \dim_{\mathbb{F}} H_i(M; \mathbb{F}).$$

You're right about $Crit(M)\geq LS(M) \geq CL(M)$. You can check the proofs in this book, which is the main reference for the LS category.

It is also true that $Crit(M)\geq BS(M) \geq CL(M)$, taking coefficients in a field for $CL(M)$. The first inequality comes from Morse theory while the latter is due to the fact that each subproduct of the largest cup product must live in some cohomology group.

There is no relation between $LS(M)$ and $BS(M)$. Consider the torus $M=T$. Then $BS(M)=4$ but $LS(M)=2$. Now consider the real projective space $M=\mathbb{R}P^n$. Then $LS(M)=n$ but $BS(M)=1$ if $n$ is even and $BS(M)=2$ if $n$ is odd.

YM78
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