Preimage by homotopy family of smooth functions is invariant under homeomorphism

Question

Suppose $f_t:M^n\rightarrow \mathbb{R}^n$ is a family of smooth functions depending smoothly on $t\in[0,1]$, where $M$ is a manifold. I would like to know when the preimage of $0$ (assume not empty) by $f_0$ is homeomorphic to $f_1^{-1}(0)$. I think requiring that the differential of each $f_t$ is surjective (or of constant rank) is enough, as the preimage will be a smooth manifold of dimension $m-n$ for each $t$. Is this correct?

Answer 1

It is not true. Let $M^n = \mathbb{R}^{n-1} \times (-1,1)$ and $F : M^n \times \mathbb{R} \to \mathbb{R}^n, F(x,s,t) = (x,s+t)$. Then the $f_t(x,s) = F(x,s,t)$ form a family as desired. But you have $f_0^{-1}(0) = \{ 0 \}$, $f_1^{-1}(0) = \emptyset$.