# Concordance implies isotopy for smooth structures on 3-manifolds?

## 1 Problem

A $3$$== Problem == ; A -manifold has a smooth structure which is essentially unique. One question is what does "essentially unique" mean. In high dimensions, smooth structures can be classified either up to concordance or up to isotopy. In fact, in high dimensions concordance implies isotopy. Let M^n be a topological manifold and let W^{n+1} be a smooth manifold homeomorphic to M \times I, one can ask if that smooth manifold W is diffeomorphic to a product, W \xrightarrow{diffeo} \partial_+ W \times I This "concordance implies isotopy" theorem (see {{cite|Kirby&Siebenmann1977}}) is: * true in high dimensions, n \geq 5. Given a topological manifold M^n, smooth structures can be classified 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 equivalence 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 the topological manifold M. \Sigma \xrightarrow{homeo} M. Two smoothings \Sigma_0 \xrightarrow{homeo} M^n and \Sigma_1 \xrightarrow{homeo} M are concordant if there is a smooth manifold W^{n+1} and a homeomorphism to M \times I that restricts to these two smoothings. 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. {{beginthm|Theorem|}} If dim M \geq 5, then concordant structures are isotopic (and hence diffeomorphic). {{endthm}} * false for simply-connected -manifolds by the failure of the h-cobordism theorem. Cappell and Shaneson proved in the non-simply connected case that there are counterexamples to the h-cobordism theorem (See {{cite|Cappel&Shaneson1987}} and {{cite|Cappell&Shaneson1987a}}). * we don't know the answer for n=3. ''Question'': Does concordance imply isotopy for smooth structures on -manifolds? This question was posed by Jim Davis at the [[:Category:MATRIX 2019 Interactions|MATRIX meeting on Interactions between high and low dimensional topology.]] == References == {{#RefList:}} [[Category:Problems]] [[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, smooth structures can be classified either up to concordance or up to isotopy. In fact, in high dimensions concordance implies isotopy.

Let $M^n$$M^n$ be a topological manifold and let $W^{n+1}$$W^{n+1}$ be a smooth manifold homeomorphic to $M \times I$$M \times I$, one can ask if that smooth manifold $W$$W$ is diffeomorphic to a product,

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

This "concordance implies isotopy" theorem (see [Kirby&Siebenmann1977]) is:

• true in high dimensions, $n \geq 5$$n \geq 5$.

Given a topological manifold $M^n$$M^n$, smooth structures can be classified 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)$$\mathcal{S}^{TOP/O}(M^n)$ is the set of equivalence classes of smoothings on $M^n$$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$$\Sigma$ to the topological manifold $M$$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 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.

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$,

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

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).

• false for simply-connected $4$$4$-manifolds by the failure of the $h$$h$-cobordism theorem. Cappell and Shaneson proved in the non-simply connected case that there are counterexamples to the $h$$h$-cobordism theorem (See [Cappel&Shaneson1987] and [Cappell&Shaneson1987a]).
• we don't know the answer for $n=3$$n=3$.

Question: Does concordance imply isotopy for smooth structures on $3$$3$-manifolds?

This question was posed by Jim Davis at the MATRIX meeting on Interactions between high and low dimensional topology.