Let $f\colon M \to N$ and $g\colon A \to N$ be two smooth maps between manifolds $A,M,N$.

Can one perturb $f$ to be transverse to $g$ (without touching $g$)?

Transverse meaning: For every $y\in f(M)\cap g(A)$ and every $x\in f^{-1}(y)$ and $a\in g^{-1}(y)$, we have $$ Df(T_x M) + Dg(T_a A) = TyN. $$

I always assumed that this was true, but searching for references, I found exercise 14 in Section 3.2 of the book by Hirsch, "Differential Topology" stating: "Is it true (as seems likely) that the set $\{f\in C^\infty(M;N): \text{$f\cap g$ transversely}\}$ is residual in $C^\infty(M;N)$ and open if $g$ is proper?"

The exercise is marked with three stars meaning "Three-star 'exercises' are problems to which I do not know the answer."

Let $f\colon M \to N$ and $g\colon A \to N$ be two smooth maps between manifolds $A,M,N$.

Can one perturb $f$ to be transverse to $g$ (without touching $g$)?

Transverse meaning: For every $y\in f(M)\cap g(A)$ and every $x\in f^{-1}(y)$ and $a\in g^{-1}(y)$, we have $$ Df(T_x M) + Dg(T_a A) = T_yN. $$

I always assumed that this was true, but searching for references, I found exercise 14 in Section 3.2 of the book by Hirsch, "Differential Topology" stating: "Is it true (as seems likely) that the set $\{f\in C^\infty(M;N): \text{$f\cap g$ transversely}\}$ is residual in $C^\infty(M;N)$ and open if $g$ is proper?"

The exercise is marked with three stars meaning "Three-star 'exercises' are problems to which I do not know the answer."