During a lecture I was given a bunch of easy propositions entitled as "observations" by the lecturer. But one of them seems more difficult and I have absolutely no idea how to "observe" it...
I'd be grateful for any (possibly online) reference, hint or hopefully a solution. The theorem states as follows:
Assumptions: Let $f:(0-\varepsilon,1+\varepsilon)\times M \to N$ be a smooth homotopy between two manifolds $M,N$. Fix $V$ - a submanifold of $N$ and assume that for all $t$ function $f_t$ is transversal to $V$.
Conclusion: There exists a smooth isotopy $\varphi$ (i.e. $\varphi_t$ is a diffeo $\forall t$) starting from $id_M$ such that $\varphi_t(f^{-1}_0(V)) = f_t^{-1}(V)$.
Update: The theorem is easy if $V$ is a point. Then we have that $f_t$ is locally a submersion and we can arrange a vector field $v_t$ (depending on $t$) with "flow" $\varphi$ such that ${d \over dt}f_t(\varphi_t(x))=0$. But I don't know if proof in the general case can follow the same scheme. It should also be ok if $f_t$ is locally a submersion [and $V$ is generic], but there is a problem if ${f_t}_*(x)({d\over dt})\nsubseteq {f_t}_*(x)(T_xM)$.
The proof may be generalized from the special case, but the result is weaker: the composition $f_t\circ \varphi_t$ is not constant, but its image stays in $V$.
It is enough to arrange a vector field $v_t$ such that ${f_t}_*(v_t + {d\over dt})\in TV$. We can do it, because ${f_t}_*(TM)+TV = TN$ (first locally and than glue with some partition of unity).
Update. We should note that the claim is true only locally (maybe also in the compact case?). Imagine $M=(0,1)\def\R{\mathbb R} $, $N=\mathbb R^3$, $V=\R^2$. Let $f_t(s)=(0,0,-{3\over 2}+s+10t)$. Then the intersection $f_t(M)\cap V$ is sometimes empty and sometimes a singleton, so there is no isotopy of $M$ preserving the inverse image of $V$.