Concordance implies isotopy for smooth structures on 3-manifolds?
CarmenRovi (Talk | contribs) |
CarmenRovi (Talk | contribs) |
||
Line 3: | Line 3: | ||
− | + | A $3$-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 $M^n$ and a smooth manifold $W^{n+1}$ homeomorphic to $M \times I$, you can ask if that smooth manifold $W$ is diffeomorphic to a product, | ||
+ | $$W \xrightarrow{diffeo} \partial_+ W \times I$$ | ||
+ | |||
+ | |||
+ | This ``concordance implies isotopy" theorem of Kirby-Siebenmann and Hirsch is: | ||
+ | \begin{itemize} | ||
+ | \item true in high dimensions, $n \geq 5$. | ||
+ | |||
+ | If you have a topological manifold $M^n$, you can classify smooth structures as homotopy classes of maps | ||
+ | $$\mathcal{S}^{TOP/O}(M^n) = [M^n, TOP/O]$$ | ||
+ | The structure set $\mathcal{S}^{TOP/O}(M^n)$ is the set of classes of smoothings on $M^n$. 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 $\Sigma$ to $M.$ | ||
+ | $$\Sigma \xrightarrow{homeo} M.$$ | ||
+ | |||
+ | Two smoothings $\Sigma_0 \xrightarrow{homeo} M^n$ and $\Sigma_1 \xrightarrow{homeo} M$ are \textit{concordant} if there is a smooth manifold $W^{n+1}$ and a homeomorphism to $M \times I$ that restricts to these two smoothings. | ||
+ | |||
+ | \medskip | ||
+ | |||
+ | Two smoothings $\Sigma_0 \xrightarrow{homeo} M$ and $\Sigma_1 \xrightarrow{homeo} M$ are isotopic if there is a smooth manifold $\Sigma$ and a level-preserving homeomorphism $H: \Sigma \times I \to M \times I$, | ||
+ | $$(H(s, t) = (F(s, t), t))$$ | ||
+ | inducing $\Sigma_i \xrightarrow{homeo} M \times \{ i \}$ for $i = 0, 1.$ | ||
+ | |||
+ | |||
+ | |||
+ | \begin{theorem} If dim $M \geq 5$, then concordant structures are isotopic (and hence diffeomorphic). | ||
+ | \end{theorem} | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | \item false for simply-connected $4$-manifolds by the failure of the $h$-cobordism theorem. Cappell and Shanneson proved in the non-simply connected case that there are counterexamples to the $h$-cobordism theorem. | ||
+ | \item we don't know the answer for $n=3$. | ||
+ | \end{itemize} | ||
+ | |||
+ | aa | ||
</wikitex> | </wikitex> |
Revision as of 06:29, 8 January 2019
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