# Existence of a map homotopic to an odd degree map which is transversal to transversal submanifolds?

my question is as follows:

Given a map $f:X\rightarrow Y$ between compact spaces of equivalent dimension, and two sub-manifolds $Z_1,Z_2 \subset Y$ which transversely intersect with $I_2[Z_1,Z_2]=1$, show that there exists a map $g:X\rightarrow Y$ which is homotopic to $f$, transversal to $Z_1$ and $Z_2$ and to $Z_1\cap Z_2$ satisfying $I_2[g^-1(Z_1),g^-1(Z_2)]=1$

I've been working on it for a while but I've not made a ton of progress. It strikes me that one can apply the Homotopy Transversality theorem from Guilleman and Pollack pg 70 to construct a map $g$ homotopic to $f$ and transversal to any of the three submanifolds the problem requires, but not to all three simultaneously; I think the approach is probably to use this theorem to construct a homotopic map $g:X\rightarrow Y$ which is transversal to say, $Z_1$ and then show that because of the condition on the intersection modulo 2, this implies the image is also transversal to $Z_2$ and the intersection; but just working through examples I haven't figured out a method of proof.

Will upvote as always, thanks in advance!

If you understand the proofs of those theorems, you should understand that you can obtain transversality to a finite number of submanifolds. To see why, look at the Corollary on p. 40. The bad values of $s$ are a set of measure zero, and the union of a finite number of sets of measure zero is another set of measure zero.