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I was reading this article about the disproof of Triangulation conjecture: it says that A. Casson disproved this conjecture in dimension 4 in the '80s

In 1982, Michael Freedman, then at the University of California, San Diego, constructed four-dimensional manifolds that didn’t allow for a natural kind of triangulation, an accomplishment that helped propel him to a Fields Medal. A few years later, Andrew Casson of Yale University proved that these particular manifolds couldn’t be triangulated at all. Yet Freedman’s and Casson’s work didn’t reveal whether triangulation is possible for all manifolds in five or more dimensions.

I'd like to know the reference for Casson's result about the existence of 4-manifolds that are not triangulable.

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  • $\begingroup$ Should post this @ MO to get more attention and answers. $\endgroup$
    – DeepSea
    Feb 23, 2015 at 8:46
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    $\begingroup$ See the bottom email here. $\endgroup$
    – user98602
    Feb 23, 2015 at 9:31
  • $\begingroup$ @MikeMiller Thank you! I write here the reference in case someone else is interested: Akbulut-McCarthy: Casson's invariant for oriented homology 3-spheres, Princeton Math. Notes 36 (1990) $\endgroup$
    – Dario
    Feb 23, 2015 at 9:52

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The bottom email here describes precisely what Casson did, and provides a source for reading about it: Akbulut-McCarthy, Casson's invariant for oriented homology 3-spheres.

As a note, it is my impression that the result was first found in a seminar, and was not written down and published for some time after that, though Taubes found a different proof in 1986 (see Gauge theory on asymptotically periodic 4-manifolds, ).

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An interesting related remark is that in dimension $4$ being triangulable is the same as admitting a PL structure (and hence as being smoothable).

I found it reviewed in this preprint, in particular there is this theorem:

Theorem 1.12 (Cannon [Can79], Edwards [Edw78, Edw06]). Let $C$ be any homology-manifold. The following are equivalent:

(i) $C$ is a manifold;

(ii) for every vertex $v$ of $C$, $\mbox{Link}(v, C)$ is simply connected.

This, together with the result that every triangulation of a manifold is a homology-manifold (also cited in the paper), and Poincaré conjecture implies that (see Theorem 1.9)

All triangulations of any $4-$manifold are PL.

Hence every non-smoothable $4-$manifold is also non-triagulable.

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