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.^[1][2][3]

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

1. Robert D. Edwards, "Suspensions of homology spheres" (2006) ArXiv (reprint of private, unpublished manuscripts from the 1970's)
2. R.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.
3. J.W. Cannon, "Σ² H³ = S⁵ / G", Rocky Mountain J. Math. (1978) 8, pp. 527-532.

• 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
• Steve Ferry, Geometric Topology Notes (See Chapter 26, page 166)