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.