# When are pullbacks of simple homotopy equivalences still simple homotopy equivalences?

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.

• You didn't define $f'$. I would suggest you attempt to reduce your question to one about simple maps, as in Definition 1.1.5 of folk.uio.no/rognes/papers/plmf.pdf, and then apply Proposition 2.1.3(c) of the same reference. – skupers Sep 8 '19 at 18:23
• I'm using the unfortunate but common notation that $f'$ is the pullback of $f$ along $g$. Translating the problem into one about simple maps sounds like a fruitful approach, thank you! – Joe Berner Sep 8 '19 at 18:37
• Before you can ask if $f'$ is a simple equivalence, you have to know that the homotopy pullback $X \times_Y Z$ is equivalent to a finite CW-complex in a preferred way. Given that under the given hypotheses it need not be equivalent to a finite CW-complex at all, it seems unlikely that there are reasonable conditions that guarantee this. – Oscar Randal-Williams Sep 9 '19 at 6:48
• Though not your question, if it suffices to work in a (reasonably nice) simplicial or PL category, you have that maps with contractible point inverses are simple and obviously the pullback along any map will give a map with contractible point inverses. – Connor Malin Aug 1 at 2:08