Double suspension theorem explained

In geometric topology, the double suspension theorem of James W. Cannon and Robert 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 Robert D. Edwards, "Suspensions of homology spheres" (2006) ArXiv (reprint of private, unpublished manuscripts from the 1970's)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. 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.
3. James W. Cannon, "Σ² H³ = S⁵ / G", Rocky Mountain J. Math. (1978) 8, pp. 527-532.

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