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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."

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  • $\begingroup$ 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. $\endgroup$
    – Klaus Niederkrüger
    Commented Jan 13, 2019 at 12:29
  • $\begingroup$ 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 $C^{\infty}(M,N)$?? $\endgroup$
    – Praphulla Koushik
    Commented Jan 13, 2019 at 13:24

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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.

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  • $\begingroup$ By the multijet transversality theorem... $\endgroup$
    – John Klein
    Commented Jan 13, 2019 at 15:39

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