Concordance implies isotopy for smooth structures on 3-manifolds?
CarmenRovi (Talk | contribs) |
CarmenRovi (Talk | contribs) (→Problem) |
||
Line 8: | Line 8: | ||
$$W \xrightarrow{diffeo} \partial_+ W \times I$$ | $$W \xrightarrow{diffeo} \partial_+ W \times I$$ | ||
− | This "concordance implies isotopy" theorem | + | This "concordance implies isotopy" theorem (see {{cite|Kirby&Siebenmann1977}}) is: |
− | + | * true in high dimensions, $n \geq 5$. | |
− | + | ||
− | + | Given a topological manifold $M^n$, you can classify smooth structures as homotopy classes of maps | |
$$\mathcal{S}^{TOP/O}(M^n) = [M^n, TOP/O]$$ | $$\mathcal{S}^{TOP/O}(M^n) = [M^n, TOP/O]$$ | ||
− | 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 | + | 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 | + | A smoothing is a homeomorphism from a smooth manifold $\Sigma$ to the topological manifold $M$. |
$$\Sigma \xrightarrow{homeo} M.$$ | $$\Sigma \xrightarrow{homeo} M.$$ | ||
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 $\Sigma_0 \xrightarrow{homeo} M^n$ 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 $H: \Sigma \times I \to M \times I$, | 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 $H: \Sigma \times I \to M \times I$, | ||
Line 27: | Line 24: | ||
inducing $\Sigma_i \xrightarrow{homeo} M \times \{ i \}$ for $i = 0, 1.$ | 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 $4$-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. | ||
+ | * we don't know the answer for $n=3$. | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
</wikitex> | </wikitex> |
Revision as of 09:28, 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, smooth structures can be classified either up to concordance or up to isotopy. In high dimensions concordance implies isotopy.
Let be a topological manifold and let be a smooth manifold homeomorphic to , one can ask if that smooth manifold is diffeomorphic to a product,
This "concordance implies isotopy" theorem (see [Kirby&Siebenmann1977]) is:
- true in high dimensions, .
Given 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 the topological manifold .
Two smoothings and are concordant if there is a smooth manifold and a homeomorphism to that restricts to these two smoothings.
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).
- false for simply-connected -manifolds by the failure of the -cobordism theorem. Cappell and Shaneson proved in the non-simply connected case that there are counterexamples to the -cobordism theorem.
- we don't know the answer for .
2 References
- [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004