OBSTRUCTIONS TO IMPOSING DIFFERENTIABLE STRUCTURES,
Abstract
Previously developed techniques are applied to this problem. Roughly, a Brouwer triangulation of topological n-manifold, M, is assumed and imbeddings are taken as a first try at coo1dinate systems covering M. These do not overlap differentiably, but an attempt is made to ''smooth them out'' so that they will. Obstructions to this smoothing are encountered, which appear in the infinite homology, with twisted coefficients in the non-orientable case. In special cases (e.g., if M is contractible) all these homology groups vanish, and the techniques suffice to construct a differentiable structure on M. A list of such cases is given. To strengthen the conclusion of the problem, the differentiable structure obtained should be compatible with the given triangulation, or some subdivision of it. The differentiable structures constructed do not have this property, for they may possess conical points. The differentiable structures obtained may be modified so as to satisfy the compatibility conditions, but it remains to be seen whether this is more than a guess. The results of R. Thom, who has also constructed an obstruction theory to attack this problem, are said to be stronger than those which appear in the present paper. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1943
- Accession Number
- AD0438546
Entities
People
- James Nunkres
Organizations
- Princeton University