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.