Suspensions of homology spheres

TL;DR

This paper proves key cases of the Double Suspension Conjecture, showing that double suspensions of homology spheres are spheres, advancing understanding of manifold topology and the structure of homology spheres.

Contribution

It provides the first solutions to the Double Suspension Conjecture for homology spheres, demonstrating that double suspensions yield spheres and thus resolving longstanding topological questions.

Findings

Double suspension of Mazur's homology 3-sphere is a 5-sphere

Double suspension of homology n-sphere bounding a contractible (n+1)-manifold is an (n+2)-sphere

Triple suspension of any homology 3-sphere is a 6-sphere

Abstract

This article is one of three highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's but never published. It presents the initial solutions of the fabled Double Suspension Conjecture. (The other two articles are: 'Approximating certain cell-like maps by homeomorphisms' and 'Topological regular neighborhoods') The manuscripts of these three articles have circulated privately since their creation. The organizers of the Workshops in Geometric Topology (http://www.math.oregonstate.edu/~topology/workshop.htm) with the support of the National Science Foundation have facilitated the preparation of electronic versions of these articles to make them publicly available. The second and third articles are still in preparation. The current article contains four major theorems: I. The double suspension of Mazur's homology 3-sphere is a 5-sphere, II. The double suspension of any homology n-sphere that bounds a contractible (n+1)-manifold is an (n+2)-sphere, III. The double suspension of any homology 3-sphere is the cell-like image of a 5-sphere. IV. The triple suspension of any homology 3-sphere is a 6-sphere. Edwards' proof of I. was the first evidence that the suspension process could transform a non-simply connected manifold into a sphere, thereby answering a question that had puzzled topologists since the mid-1950's if not earlier. Results II, III and IV represent significant advances toward resolving the general double suspension conjecture: the double suspension of every homology n-sphere is an (n+2)-sphere. [That conjecture was subsequently proved by J. W. Cannon (Annals of Math. 110 (1979), 83-112).]