Let $f : X \to Y$ and $g : Z \to Y$ be continuous maps between finite CW complexes. If $f$ is a simple homotopy equivalence, are there conditions on $g$ which guarantee that its pullback $f'$ is a simple homotopy equivalence?
Since homotopy equivalences themselves are not even preserved under pullback, a natural condition is for $f$ or $g$ to be a Serre fibration. The assumption that $f$ is a Serre fibration is better for my purposes, so let's assume that.
A possible reduction of the problem goes like this: Assume $f$ is a Serre fibration, then the pullback is a homotopy pullback, and we can replace $f$ with a formal deformation (using the terminology in Cohen's book), that is $f = f_0 \circ f_1 \circ \ldots f_n$ where each $f_i$ is an elementary expansion or contraction. We can then reduce the question to when the pullback of an elementary expansion/contraction is itself a simple homotopy equivalence. We can write down a ladder of pullbacks of the $f_i$s factoring $f'$, and if each $f_i'$ is a simple homotopy equivalence then the composition of them is too. I'm not sure how much this reduction really helps though.