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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.

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    $\begingroup$ 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. $\endgroup$
    – Ryan Budney
    Commented Jul 14, 2021 at 7:27
  • $\begingroup$ OK, but would it work for a (relative) cobordism between 3-manifolds with boundary? (which is the case I'm really interested in using) $\endgroup$
    – Ying Hong Tham
    Commented Jul 14, 2021 at 16:45

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PL manifolds are triangulable. The theorem you want is Pachner's theorem, that any two triangulations are related by stellar moves.

You can turn triangulations into special PL handle decompositions, and PL handle decompositions you can triangulate.

From this perspective, you could reduce your problem to showing that PL-triangulations could be viewed as very specific PL-handle decompositions. Then you argue that Pachner moves are obtainable by elementary handle operations, to complete the correspondence between handle decompositions up to elementary operations and triangulations up to stellar moves.

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