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 you can classify smooth structures either up to concordance or up to isotopy. In high dimensions concordance implies isotopy.
If you have a topological manifold and a smooth manifold homeomorphic to , you 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 \textit{concordant} if there is a smooth manifold and a homeomorphism to that restricts to these two smoothings.
\medskip
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}
aa