Concordance implies isotopy for smooth structures on 3-manifolds?
1 Problem
A -manifold has a smooth structure which is essentially unique. One question is what does "essentially unique" mean.
In high dimensions, smooth structures can be classified either up to concordance or up to isotopy. In high dimensions concordance implies isotopy.
Let be a topological manifold and let be a smooth manifold homeomorphic to , one can ask if that smooth manifold is diffeomorphic to a product,
This "concordance implies isotopy" theorem of Kirby-Siebenmann and Hirsch is: \begin{itemize} \item true in high dimensions, .
If you have a topological manifold , you can classify smooth structures as homotopy classes of maps
The structure set is the set of classes of smoothings on . But it is important to decide what is meant by the ``structure" and there are two possible equivalence relations: concordance or isotopy.
A smoothing is a homeomorphism from a smooth manifold to
Two smoothings and are concordant if there is a smooth manifold and a homeomorphism to that restricts to these two smoothings.
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Two smoothings and are isotopic if there is a smooth manifold and a level-preserving homeomorphism ,
inducing for
Theorem 1.1. If dim , then concordant structures are isotopic (and hence diffeomorphic).
\item false for simply-connected -manifolds by the failure of the -cobordism theorem. Cappell and Shanneson proved in the non-simply connected case that there are counterexamples to the -cobordism theorem. \item we don't know the answer for . \end{itemize}
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