# perturbing one map to be transverse to a second map

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."

• Don't know if the general situation can be solved, but in some particular cases, one could perturb f(M) to be a submanifold of N and suppose its normal bundle is trivial, then composing g with the projection to the fiber and using Sard's theorem, we would get that most sections in this normal bundle are transverse to g. Jan 13, 2019 at 12:29
• May be you should add your comment in the question... This is interesting question.... What does it mean to say a set is residual in $C^{\infty}(M,N)$?? Jan 13, 2019 at 13:24

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.