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Connor Malin
Connor Malin
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For $n>4$ one may identify $\pi_n(Top/O)$ via smoothing theory as concordance class of smooth structures on $S^n$ where two structures are concordant if they bound a smooth structure on the product with an interval. I have seen it claimed that these homotopy groups are finite, for example when it is asserted $BTop \rightarrow BO$$BO \rightarrow BTop$ is a rational equivalence.

One would like to use Kervaire and Milnor's work which shows the finiteness of oriented homotopy spheres, but I do not see how to go from a statement about diffeomorphism type to a statement about concordance class. Is it true for a sphere concordant is equivalent to diffeomorphic? That concordance implies diffeomorphism is true by the h-cobordism theorem, but I believe for some manifolds at least the other way does not hold.

For $n>4$ one may identify $\pi_n(Top/O)$ via smoothing theory as concordance class of smooth structures on $S^n$ where two structures are concordant if they bound a smooth structure on the product with an interval. I have seen it claimed that these homotopy groups are finite, for example when it is asserted $BTop \rightarrow BO$ is a rational equivalence.

One would like to use Kervaire and Milnor's work which shows the finiteness of oriented homotopy spheres, but I do not see how to go from a statement about diffeomorphism type to a statement about concordance class. Is it true for a sphere concordant is equivalent to diffeomorphic? That concordance implies diffeomorphism is true by the h-cobordism theorem, but I believe for some manifolds at least the other way does not hold.

For $n>4$ one may identify $\pi_n(Top/O)$ via smoothing theory as concordance class of smooth structures on $S^n$ where two structures are concordant if they bound a smooth structure on the product with an interval. I have seen it claimed that these homotopy groups are finite, for example when it is asserted $BO \rightarrow BTop$ is a rational equivalence.

One would like to use Kervaire and Milnor's work which shows the finiteness of oriented homotopy spheres, but I do not see how to go from a statement about diffeomorphism type to a statement about concordance class. Is it true for a sphere concordant is equivalent to diffeomorphic? That concordance implies diffeomorphism is true by the h-cobordism theorem, but I believe for some manifolds at least the other way does not hold.

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Connor Malin
Connor Malin
  • 5.8k
  • 1
  • 14
  • 37

For $n>4$ one may identify $\pi_n(Top/O)$ via smoothing theory as concordance class of smooth structures on $S^n$ (perhaps with an orientation?), where two structures are concordant if they bound a smooth structure on the product with an interval. I have seen it claimed that these homotopy groups are finite, for example when it is asserted $BTop \rightarrow BO$ is a rational equivalence.

One would like to use Kervaire and Milnor's work which shows the finiteness of oriented homotopy spheres, but I do not see how to go from a statement about diffeomorphism type to a statement about concordance class. Is it true for a sphere concordant is equivalent to diffeomorphic? That concordance implies diffeomorphism is true by the h-cobordism theorem, but I believe for some manifolds at least the other way does not hold.

For $n>4$ one may identify $\pi_n(Top/O)$ via smoothing theory as concordance class of smooth structures on $S^n$ (perhaps with an orientation?), where two structures are concordant if they bound a smooth structure on the product with an interval. I have seen it claimed that these homotopy groups are finite, for example when it is asserted $BTop \rightarrow BO$ is a rational equivalence.

One would like to use Kervaire and Milnor's work which shows the finiteness of oriented homotopy spheres, but I do not see how to go from a statement about diffeomorphism type to a statement about concordance class. Is it true for a sphere concordant is equivalent to diffeomorphic? That concordance implies diffeomorphism is true by the h-cobordism theorem, but I believe for some manifolds at least the other way does not hold.

For $n>4$ one may identify $\pi_n(Top/O)$ via smoothing theory as concordance class of smooth structures on $S^n$ where two structures are concordant if they bound a smooth structure on the product with an interval. I have seen it claimed that these homotopy groups are finite, for example when it is asserted $BTop \rightarrow BO$ is a rational equivalence.

One would like to use Kervaire and Milnor's work which shows the finiteness of oriented homotopy spheres, but I do not see how to go from a statement about diffeomorphism type to a statement about concordance class. Is it true for a sphere concordant is equivalent to diffeomorphic? That concordance implies diffeomorphism is true by the h-cobordism theorem, but I believe for some manifolds at least the other way does not hold.

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Connor Malin
Connor Malin
  • 5.8k
  • 1
  • 14
  • 37

Finiteness of $\pi_n(Top/O)$

For $n>4$ one may identify $\pi_n(Top/O)$ via smoothing theory as concordance class of smooth structures on $S^n$ (perhaps with an orientation?), where two structures are concordant if they bound a smooth structure on the product with an interval. I have seen it claimed that these homotopy groups are finite, for example when it is asserted $BTop \rightarrow BO$ is a rational equivalence.

One would like to use Kervaire and Milnor's work which shows the finiteness of oriented homotopy spheres, but I do not see how to go from a statement about diffeomorphism type to a statement about concordance class. Is it true for a sphere concordant is equivalent to diffeomorphic? That concordance implies diffeomorphism is true by the h-cobordism theorem, but I believe for some manifolds at least the other way does not hold.