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It is well known that in dimension $4$, the notion of piecewise linear manifolds and the notion of smooth manifolds are the same [1][2]. On the other hand, the computations of Donaldson invariants involve moduli spaces constructed from the smooth structure of the underlying manifold. However, if smooth $=$ PL then there must be a translation of it into the piecewise linear world.

What's the translated result? Please provide some references if it is still under work.

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    $\begingroup$ As far as I am aware, this has not been done. I have a more elementary project of building algorithms to compute Rochlin invariants of 3-manifolds purely in the language of triangulations. So this is a very simple smooth 4-manifold theory invariant, done in the language of triangulations. In principle this is "done" although it is not implemented on a computer, as of yet. $\endgroup$
    – Ryan Budney
    Commented Aug 11, 2021 at 19:22
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    $\begingroup$ With gauge theory type invariants, the efficient approach would not be to fully reproduce the defining mechanisms of the invariant as they are not "finitary enough" for computer implementation. What you need to do is find new ways of looking at these invariants, so they are algorithmically computable. I suppose I should clarify, is your interest in computation, or simply translating the category of definition? The latter has a tautological answer: smooth the manifold then compute. $\endgroup$
    – Ryan Budney
    Commented Aug 11, 2021 at 19:35
  • $\begingroup$ Yes, I do hope to obtain an algorithm that will halt in finite time for manifolds with a finite triangulation. Where can I find your work on Rochlin invariants? $\endgroup$
    – Student
    Commented Aug 11, 2021 at 20:34
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    $\begingroup$ It's not in a complete package yet. There's two results that are relevant. My paper on combinatorial spin structures on triangulated manifolds. You can find that on the arXiv. This is largely implemented in the 4-manifolds branch of the Regina software. The other main step is finding triangulations of the 4-manifolds that cobound triangulated 3-manifolds. My student Sam Churchill turned the Costantino-Thurston paper into an "in principle" algorithm in his M.Sc thesis, which I think should be available online. That still needs implementation -- hopefully in Regina in the next few years. $\endgroup$
    – Ryan Budney
    Commented Aug 11, 2021 at 20:49
  • $\begingroup$ Thanks for the pointer. One thing I find it hard is to make sense of the connections on bundles over PL manifolds, or even better, tro translate a smooth bundle with connection into the PL world. Has this been progressed? $\endgroup$
    – Student
    Commented Aug 12, 2021 at 15:06

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