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Manifolds with unique differentiable structure. (English) Zbl 0547.57025

The main result is the following Theorem: Let \(n\neq 4\). Every closed oriented smooth manifold of dimension n is bordant in the oriented bordism group \(\Omega_ n\) to one with unique differentiable structure up to diffeomorphism.
The main steps in the proof are as follows: First the notion of a type BSO manifold is introduced (defined mainly in terms of the behavior of the classifying map of the stable tangent bundle under \(\pi_ i)\). An analysis of such manifolds is made and it is shown that being of type BSO is independent of the differentiable structure, and that every oriented closed manifold of dimension \(\neq 3\) is bordant in \(\Omega_ n\) to a manifold of type BSO. Next a notion of stably diffeomorphic manifolds is introduced and it is shown that for type BSO manifolds, being stably diffeomorphic is roughly equivalent to being bordant. If then M is a manifold of type BSO and M’, M” are two differentiable structures on M then M’ and M” are bordant and thus stably diffeomorphic which, it is shown, implies M’ and M” are diffeomorphic structures.
Reviewer: M.V.Mielke

MSC:

57R55 Differentiable structures in differential topology
57R10 Smoothing in differential topology
57R75 \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism
57R65 Surgery and handlebodies
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