Concordance implies isotopy for smooth structures on 3-manifolds?

1 Problem

A $3$$== Problem == <wikitex>; A $-manifold has a smooth structure which is essentially unique. One question is what does ``essentially unique" mean. In high dimensions you can classify smooth structures either up to concordance or up to isotopy. In high dimensions concordance implies isotopy. If you have a topological manifold $M^n$ and a smooth manifold $W^{n+1}$ homeomorphic to $M \times I$, you can ask if that smooth manifold $W$ is diffeomorphic to a product, $$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 $-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> == References == {{#RefList:}} [[Category:Questions]] [[Category:Research questions]]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$$M^n$ and a smooth manifold $W^{n+1}$$W^{n+1}$ homeomorphic to $M \times I$$M \times I$, you can ask if that smooth manifold $W$$W$ is diffeomorphic to a product,

$\displaystyle W \xrightarrow{diffeo} \partial_+ W \times I$ $$\displaystyle W \xrightarrow{diffeo} \partial_+ W \times I$$

This ``concordance implies isotopy" theorem of Kirby-Siebenmann and Hirsch is: \begin{itemize} \item true in high dimensions, $n \geq 5$$n \geq 5$.

If you have a topological manifold $M^n$$M^n$, you can classify smooth structures as homotopy classes of maps

$\displaystyle \mathcal{S}^{TOP/O}(M^n) = [M^n, TOP/O]$ $$\displaystyle \mathcal{S}^{TOP/O}(M^n) = [M^n, TOP/O]$$

The structure set $\mathcal{S}^{TOP/O}(M^n)$$\mathcal{S}^{TOP/O}(M^n)$ is the set of classes of smoothings on $M^n$$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$$\Sigma$ to $M.$$M.$

$\displaystyle \Sigma \xrightarrow{homeo} M.$ $$\displaystyle \Sigma \xrightarrow{homeo} M.$$

Two smoothings $\Sigma_0 \xrightarrow{homeo} M^n$$\Sigma_0 \xrightarrow{homeo} M^n$ and $\Sigma_1 \xrightarrow{homeo} M$$\Sigma_1 \xrightarrow{homeo} M$ are \textit{concordant} if there is a smooth manifold $W^{n+1}$$W^{n+1}$ and a homeomorphism to $M \times I$$M \times I$ that restricts to these two smoothings.

\medskip

Two smoothings $\Sigma_0 \xrightarrow{homeo} M$$\Sigma_0 \xrightarrow{homeo} M$ and $\Sigma_1 \xrightarrow{homeo} M$$\Sigma_1 \xrightarrow{homeo} M$ are isotopic if there is a smooth manifold $\Sigma$$\Sigma$ and a level-preserving homeomorphism $H: \Sigma \times I \to M \times I$$H: \Sigma \times I \to M \times I$,

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

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

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

\item false for simply-connected $4$$4$-manifolds by the failure of the $h$$h$-cobordism theorem. Cappell and Shanneson proved in the non-simply connected case that there are counterexamples to the $h$$h$-cobordism theorem. \item we don't know the answer for $n=3$$n=3$. \end{itemize}

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