Analog of Cerf theory in PL

Is there an analog of Cerf theory in PL?

More specifically, given two handle decompositions of a PL (relative) cobordism $$W$$, is it always possible to go from one handle decomposition to the other via a sequence of handle slides and handle cancellations?

I think I have an argument, but I wanted to know if it is already known (and also check my argument): Choose some smoothing of the cobordism $$W$$. Construct Morse functions $$f_0$$ and $$f_1$$ that give the two handle decompositions (but smoothed). Find a homotopy $$f_t$$ that interpolates them such that $$f_t$$ only has at worst birth-death singularities - then $$f_t$$ gives a corresponding set of moves between handle decompositions. Approximate $$W \times I$$ by a triangulation, and approximate $$f_t$$ by a PL map. This should give the sequence of handle moves in PL.

• This does not seem like an argument. For example, what if the manifold is not smoothable? Even if it's smoothable, your PL handle decomposition might not be smoothable in any useful sense. Cerf theory by its nature is a smooth theory. It sounds like you are interested in less than proper Cerf theory. You want a tool for understanding how various PL handle decompositions of a manifold are related. Jul 14, 2021 at 7:27
• OK, but would it work for a (relative) cobordism between 3-manifolds with boundary? (which is the case I'm really interested in using) Jul 14, 2021 at 16:45