In the 1970’s Robert D. Edwards wrote, but did not publish, the following three highly influential articles on the topology of manifolds.
- Suspensions of homology spheres
- Approximating certain cell-like maps by homeomorphisms
- Topological regular neighborhoods
The first of these presents the initial solutions of the fabled Double Suspension Conjecture. The second gives the definitive manifold recognition theorem for identifying manifolds among resolvable homology manifolds. The third develops a comprehensive theory of regular neighborhoods of locally flatly embedded topological manifolds in high dimensional topological manifolds. The manuscripts of these three articles have circulated privately since their creation. The organizers of the Workshops in Geometric Topology with the support of the National Science Foundation have facilitated the preparation of electronic versions of these articles to make them publicly available.
ARTICLE #1: Suspensions of homology spheres (download in PDF)
This 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. [The general conjecture was subsequently proved by J. W. Cannon (Annals of Math. 110 (1979), 83-112).]
ARTICLE #2: Approximating certain maps by homeomorphisms
Work on this manuscript is still in progress.
ARTICLE #3: Topological regular neighborhoods (download in PDF)
This article contains a comprehensive development of a theory of regular neighborhoods of locally
flatly embedded topological manifolds in high dimensional topological manifolds.