In geometric topology, the double suspension theorem of James W. Cannon (Cannon (1979)) and Robert D. Edwards states that the double suspension S2X of a homology sphere X is a topological sphere. 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.
See also Disjoint discs property
References
Cannon, James W. (1979), "Shrinking cell-like decompositions of manifolds. Codimension three", Annals of Mathematics, Second Series, 110 (1): 83–112, doi:10.2307/1971245, ISSN 0003-486X, MR 0541330 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, ISBN 978-3-540-09243-8, MR 0554220 Steve Ferry, Geometric Topology Notes (See Chapter 26, page 166)