Let $M$ be a PL-manifold that is a homotopy sphere (PL stands for Piecewise Linear). Does it follow that $M$ is PL-homeomorphic to the sphere $S^n$ (with the usual PL-structure)? Here is the background:

Zeeman (1962 [ 2 ]: The Poincaré conjecture for $n\geq 5$) writes:

[Smale] assumed a differentiable structure upon the manifold instead of a combinatorial structure, and because of the triangulation theorem for differentiable manifolds, this was at first a weaker version than Stallings' result. Then, however, he used a constructed structed differentiable structure to prove a stronger combinatorial version (which we state without proof):

``THEOREM 3 (Differential: Smale). Let $M^n$ be a connected closed combinatorial manifold, $n\ne 4,5,7$. If $M^n$ is a homotopy sphere then it is a combinatorial sphere.''

I cannot find a proof for this statement. The best I could find is the following:

Stallings (Polyhedral homotopy spheres [ 1 ]) says that it is combinatorially equivalent to $\mathbb{R}^n$ away from a point.

[ 1 ] J. Stallings. Polyhedral homotopy-spheres. Bull. Amer. Math. Soc. 66 (1960), 485-488. ProjetEuclid link.

[ 2 ] E. Zeeman. The Poincaré conjecture for $n\ge 5$. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 198-204 Prentice-Hall, Englewood Cliffs, N.J.

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    $\begingroup$ See theorem 3.12 in arxiv.org/abs/math/0105047 and references therein. $\endgroup$ – Igor Belegradek Aug 24 '19 at 13:12
  • $\begingroup$ Yes, that's a good answer. It does follow indeed from Kirby-Siebenmann. Incidentally, it also follows from K-S that every topological manifold that is h.e. to $S^n$ is homeomorphic to $S^n$, which I guess does not follow from any of the original approaches. This still begs the question though, whether K-S is necessary. Zeeman seems to suggest that it should have been known by the early 1960's. $\endgroup$ – Stefan Friedl Aug 24 '19 at 20:21
  • $\begingroup$ I don't think Kirby-Siebenmann's results are relevant here. As Rudyak says before theorem 3.12, everything was settled in the 60s. The details are not on my fingertips though. I briefly looked in Stallings' notes and could not immediately see how to use them to adapt Smale's argument to the PL setting. $\endgroup$ – Igor Belegradek Aug 25 '19 at 3:31
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    $\begingroup$ I also had a look at Stallings notes, but it is not clear how it helps. I guess many of these arguments are only comprehensible if one is immersed in the PL-language, which people were in the 60s. This teaches us again the lesson that results should be written up in such a way that they are comprehensible by future generations who might have a very different background. $\endgroup$ – Stefan Friedl Aug 25 '19 at 7:25

For spheres of dimension $n>5$ the PL Poincare conjecture follows from the s-cobordism theorem. Indeed, removing disjoint two small open disks one gets an s-cobordism (this uses excision in homology, Poincare duality and Hurewicz theorem). The s-cobordsim is PL trivial (if the cobordsim has dimension is $>5$) from where a PL homeomorphism to a sphere should be obvious. This is done in detail eg in Rourke-Sanderson PL topology book, pp 8-10.

The 5-dimensional case involves a trick. By a surgery theoretic argument Kervaire proved that every PL homology $n$-sphere $\Sigma$ with $n\neq 3$ bounds a contractible PL manifold, see p.71 in [Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc. 144 (1969), 67–72]. Removing a small open disk from the interior of that manifold gives a cobordism $W$ between $\Sigma$ and $S^5$ which is an $s$-cobordism since $\Sigma$ is simply-connected. Since $\dim(W)>5$ the s-cobordism is trivial, so $\Sigma$ is PL homeomorphic to $S^5$.

It seems that the result was first discovered through the smoothing theory as follows:

Step 1 is to show that any homotopy $5$-sphere has a smooth structure. Munkres and Hirsch developed an obstruction theory for putting a smooth structure on a PL manifold. The obstruction groups are cohomology groups of $M$ rel boundary (if any) with coefficients in the groups $\Gamma^k$ of isotopy classes of diffeomorphism of $S^{k-1}$ moduli those that extend to the $k$-disk. Now $\Gamma^k=0$ for $k\le 4$ where the hardest case $k=4$ is a famous theorem of Cerf. This shows that any PL $5$-manifold has a smooth structure.

Step 2 is to show that any smooth homotopy $5$-sphere $\Sigma$ bounds a smooth compact contractible manifold $W^6$. This is done by a surgery theoretic argument staring with some $6$-manifolds with boundary $\Sigma$ and the simplifying it by surgery. This is due to Milnor-Kevaire and Wall (independently). See theorem 9.1 of Milnor's "Lectures on the h-cobordsim theorem".

Step 3 is to remove a small open ball from the interior of $W^6$ to get a smooth $h$-cobordism between $\Sigma$ and $S^5$, which is trivial by the smooth $h$-cobordism theorem. Thus $\Sigma$ is diffeomorphic and hence PL homeomorphic to $S^5$.

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  • $\begingroup$ Thanks! I didn't know that Rourke-Sanderson does the PL h-cobordism theorem. It still feels like Zeeman was a little quick. It's nice to know that one does not need scary math (Kirby-Siebenmann) but that one can do with much better understood mathematics. $\endgroup$ – Stefan Friedl Aug 26 '19 at 19:07

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