If X is a
piecewise-linear homology sphere but not a sphere, then its double suspension S2X (with a triangulation derived by applying the double suspension operation to a triangulation of X) is an example of a triangulation of a topological sphere that is not piecewise-linear. The reason is that, unlike in piecewise-linear manifolds, the link of one of the suspension points is not a sphere.
^Robert D. Edwards, "
Suspensions of homology spheres" (2006) ArXiv (reprint of private, unpublished manuscripts from the 1970's)
^Robert D. Edwards, "The topology of manifolds and cell-like maps", Proceedings of the International Congress of Mathematicians, Helsinki, 1978 ed. O. Lehto, Acad. Sci. Fenn (1980) pp 111-127.
^James W. Cannon, "Σ2 H3 = S5 / G", Rocky Mountain J. Math. (1978) 8, pp. 527-532.
Latour, François (1979), "Double suspension d'une sphère d'homologie [d'après R. Edwards]", Séminaire Bourbaki vol. 1977/78 Exposés 507–524, Lecture Notes in Math. (in French), vol. 710, Berlin, New York:
Springer-Verlag, pp. 169–186,
doi:
10.1007/BFb0069978,
ISBN978-3-540-09243-8,
MR0554220