# Piecewise linear Poincaré conjecture

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.

• See theorem 3.12 in arxiv.org/abs/math/0105047 and references therein. Aug 24, 2019 at 13:12
• 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. Aug 24, 2019 at 20:21
• 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. Aug 25, 2019 at 3:31
• 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. Aug 25, 2019 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 $$S^5$$.

• 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. Aug 26, 2019 at 19:07