# 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 $$S^n$$ 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.

• Essay V of Kirby-Siebenmann is one reference. – skupers Oct 26 '20 at 4:06
• Thank you. For anyone interested, in my copy of their work it is on page 251. The claim is that in dimensions where smoothing theory applies, the homotopy groups are identical to the oriented homotopy spheres. When I have understood their argument I will write it up if this question has no submitted answers – Connor Malin Oct 26 '20 at 20:37
• @skupers Since you are here, I hope you won't mind if I ask you a related question I thought of while reading your notes. It seems that $Top/O$ and the block diffeomorphism groups for the disk contain the exact same information in their homotopy groups. Perhaps after looping some number of times, is it possible these spaces are homotopy equivalent? – Connor Malin Oct 26 '20 at 20:41
• Yes, see page 4 of jstor.org/stable/1997010). – skupers Oct 26 '20 at 21:22

Here I essentially am just repeating what Siebenmann wrote in Essay V of the Kirby-Siebenmann volume while fetching some definitions from early sections:

Let $$n \geq 5$$. By work of Kirby and Siebenmann on the connectivity of $$Top(n)/O(n) \rightarrow Top/O$$, $$\pi_n(Top/O)=\pi_n(Top(n)/O(n))$$. The latter is what we will choose to work with. In section 5.3 the space $$\Omega^n Top(n)/O(n)$$ was identified with the space of smoothings of $$S^n$$ relative to a disk with standard smooth structure. Hence, $$\pi_n(Top(n)/O(n))$$ is the concordance classes of such smooth structures.

We can consider the inclusion of such smoothings into the homotopy smoothings of $$S^n$$ relative to a disk (perhaps up to h-cobordism?). Surjectivity follows from the Poincare conjecture and injectivity from what Siebenmann calls "Alexander isotopy", I am not sure what this is.

Then this set of homotopy smoothings of $$S^n$$ relative to a disk has a map to the group of oriented homotopy spheres given by using the homotopy equivalence to transport the orientation of $$S^n$$ to the homotopy sphere and then forgetting the homotopy equivalence. This map is surjective since any homotopy sphere contains a standard disk in it. The main thing to showing injectivity is that up to orientation there is a single isotopy class of embedded disks in our homotopy sphere.

This chain of bijections shows that $$\pi_n(Top/O)$$ for $$n\geq 5$$ is $$\Theta_n$$, the group of oriented homotopy spheres. I think any subtlety in this proof would arise from showing that including smoothings into homotopy smoothings is injective, so I will look further into "Alexander isotopy".

• Usually the "Alexander isotopy" is a reference to the null-homotopy of the inclusions $Aut^{PL)(D^n fix \partial) \to Aut^{Top}(D^n fix \partial)$ and the same for the inclusions of diffeomorphisms of $D^n$ rel boundary in homeomorphisms. – Ryan Budney Oct 26 '20 at 23:55
• @RyanBudney Think the tex is a bit messed up. Are you saying it is related to the Alexander trick? – Connor Malin Oct 27 '20 at 0:14
• Yes, it's a space-level version of the Alexander trick. – Ryan Budney Oct 27 '20 at 0:17