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| | == Problem == | | == Problem == |
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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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| − | 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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