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Question abstract: Let $N$ be a compact smooth manifold without borders. Consider two sublevel sets of a function defined on a manifold such that the De Rham cohomology of these sublevel sets is equal to $H^*(N)$. Is it true that the minimum number of critical points of a function in such a set is achieved by functions defined on $N$ (under some technical hypothesis to ensure that the well-known relationships between the De Rham relative cohomology and the number of critical points holds) ?

Let $M$ be a smooth paracompact manifold without borders, $f\in C^{\infty}(M)$ and $f^c=\{x\in M \ | \ f(x)\leq c\}$. Assume that $(f,M)$ is Palais-Smale, which means that for every sequence $\{x_i \}$ if $\{|f(x_i)|\}$ is bounded and $\lim_{i \rightarrow \infty} f(x_i)'=0$ then there is a subsequence converging to a critical point. Let $N$ be a compact smooth manifold without borders such that $H^{*}(N)\cong H^*(f^b,f^a)$ where $b>a$, $f^{-1}(a)$ has no critical points and is $\neq \emptyset$ and $H^*$ is the de Rham relative cohomology.

Under this assumptions it is easy to see that given a non zero element $\alpha$ of $H^*(f^b, f^a)$ one has that $c(\alpha , f) = \inf \{ c \ | \ i_c^*\alpha \neq 0 \}$ is always a critical value for $f \ $ (here $i_c : f^c \rightarrow f^b$ is the inclusion and $i_c^*\alpha \in H^*(f^c, f^a) $). Of course given a function $g$ one can map non zero elements of $H^*(N)$ to critical values in the same way. Finally, the cup - lenght of $H^*(N)$ is a strict lower bound for the number of critical points of $g$ (and if I'm not mistaken the proof in the link above can be generalized to show that the the cup - lenght is also a lower bound for the number of critical points of $f$ in $f^b/f^a$).

Is it true that there always exists $g\in C^{\infty}(N)$ such that the number of critical points of $g$ is less than or equal to the number of critical points of $f$ restricted to $f^b\setminus f^a$?

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