Let $F:\mathbb{S}^{2}\times\lbrack0,1]\rightarrow\mathbb{R}$ be a smooth
($C^{\infty}$) function and $f\_{t}(x)=F(x,t)$. Suppose that $f\_{0}=f\_{1\text{
}}$is the projection over $z$-axis, so point $P=(0,0,1)$ is an absolute
maximum of both $f\_{0}$ and $f\_{1}$. Let $A\_{t}$ be the critical points set of
$f\_{t}$ and let $A=\cup\_{t}(A\_{t},t)$. My question is whether points $(P,0)$
and $(P,1)$ belong to one and the same component of set $A$? If so, this would
be some kind of stability result for the critical points set under homotopy.


If this is something well-known or a counter-example exists, any references
are welcome. (Of course, the same may be asked in a fairly more general
setting, for manifolds etc...)



Stability of the critical points set
If $f\_t$ is generic then $A$ is formed by a collection of curves in $S^2\times [0,1]$.
So, $(P,0)$ is connected to either 


1. $(P,1)$ --- this happens for the constant $f\_t$.
2. $(-P,1)$ --- this happens for the projection of a generic sphere eversion, otherwise the orientation would not change.
3. $(-P,0)$ --- It can not happen --- this follow since the number of critical points counted with singes ("$+$" for min and max and "$-$" for saddle) has to be $2$ for every $t$.


