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I would like to understand better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincaré homology sphere is homeomorphic to $S^5$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $K$ be a polyhedron and $M$ a smooth manifold. We say that a map $f \colon K \to M$ is piecewise differentiable (PD) if there exists a triangulation of $K$ such that the restriction of $f$ to each simplex is smooth. We will say that $f$ is a PD homeomorphism if $f$ is piecewise differentiable, a homeomorphism, and the restriction of $f$ to each simplex has injective differential at each point.

I would like to modify the definition. First, ask that $f$ restricts analytically to each simplex (stronger condition). Second ask that the restriction has injective differential in the interior of each simplex (weaker condition). Now, the question:

Question. Suppose $X$ is a simplicial complex that is homeomorphic to $S^5$. Suppose that the homeomorphism $\varphi: X\to S^5$ can be realized so that its restriction to the interior of each simplex in $X$ is an analytic diffeomorphism onto its image. Is it true then that $X$ is $PL$ homeomorphic to the standard sphere? (This would mean that there is a map $\varphi': X\to S^5$ that is $PL$ on each simplex of $X$ for a standard $PL$ structure on $S^5$.)

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    $\begingroup$ Yes. Actually, you need much less, for instance, Lipschitz maps will suffice. I will add a full answer sometime later, unless somebody does it before. $\endgroup$
    – Moishe Kohan
    Commented May 7, 2021 at 3:18
  • $\begingroup$ Dear Moishe, thanks for sharing this good news! Will be very interested to learn why Lipschitz maps are sufficient. (I also hope that for what you have in mind these maps are asked to be Lipschtiz only on the interior of each simplex (of each dimension), but not on closed simplices. Indeed I am thinking about analytic maps that can behave as $\sqrt x$ behaves on $[0,1]$ at 0). $\endgroup$
    – aglearner
    Commented May 7, 2021 at 8:07
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    $\begingroup$ @MoisheKohan , so far no luck with answers. If you could add even a brief one, I would be very grateful. $\endgroup$
    – aglearner
    Commented May 14, 2021 at 22:07

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