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."



perturbing one map to be transverse to a second map
I'm not sure what exactly you mean by perturb, but you can always make $f$ transverse to $g$ by a homotopy and this is often enough. This is proved in Section IV.2 of Kosinski's "Differential Manifolds" book, in particular see Corollary IV.2.5.


