Concordance implies isotopy for smooth structures on 3-manifolds?

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A $3$-manifold has a smooth structure which is essentially unique. One question is what does ``essentially unique" mean.
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In high dimensions you can classify smooth structures either up to concordance or up to isotopy. In high dimensions concordance implies isotopy.
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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,
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$$W \xrightarrow{diffeo} \partial_+ W \times I$$
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This ``concordance implies isotopy" theorem of Kirby-Siebenmann and Hirsch is:
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\begin{itemize}
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\item true in high dimensions, $n \geq 5$.
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If you have a topological manifold $M^n$, you can classify smooth structures as homotopy classes of maps
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$$\mathcal{S}^{TOP/O}(M^n) = [M^n, TOP/O]$$
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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.
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A smoothing is a homeomorphism from a smooth manifold $\Sigma$ to $M.$
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$$\Sigma \xrightarrow{homeo} M.$$
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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.
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\medskip
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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 $H: \Sigma \times I \to M \times I$,
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$$(H(s, t) = (F(s, t), t))$$
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inducing $\Sigma_i \xrightarrow{homeo} M \times \{ i \}$ for $i = 0, 1.$
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\begin{theorem} If dim $M \geq 5$, then concordant structures are isotopic (and hence diffeomorphic).
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\end{theorem}
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\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.
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\item we don't know the answer for $n=3$.
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\end{itemize}
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Revision as of 05:29, 8 January 2019

1 Problem


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,

\displaystyle 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

\displaystyle \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.

\displaystyle \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,

\displaystyle (H(s, t) = (F(s, t), t))

inducing \Sigma_i \xrightarrow{homeo} M \times \{ i \} for i = 0, 1.



Theorem 1.1. If dim M \geq 5, then concordant structures are isotopic (and hence diffeomorphic).











\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}

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2 References

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