Double suspension theorem

In geometric topology, the double suspension theorem of Cannon (1979) and R. 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.

References

• Cannon, J. 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 541330 
• 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) 710, Berlin, New York: Springer-Verlag, pp. 169–186, doi:10.1007/BFb0069978, ISBN 978-3-540-09243-8, MR 554220