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 DeRahm 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 it is also a lower bound for the critical values of $f$ in $f^b/f^a$).
Is it true in general 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 $f^b\setminus f^a$?