For a compact closed smooth manifold $X$, the group Diff(X) has a natural homomorphism $\Phi$ to the homeomorphism group Homeo(X). If $X$ has dimension at least $5$, I'm looking for some general information about the map(s) $\Phi\_\*$ induced on the homotopy groups of these spaces. I’m mainly interested in the simply connected case but of course would be interested in a more general result.


In particular, are the kernel and cokernel of $\Phi\_\*$ always finite for every homotopy group? This would be my guess, based naively on the statement that a higher-dimensional simply connected manifold has finitely many smoothings. I tried to look in some older literature (eg papers by Burghelea-Lashof) but am a bit overwhelmed by the notation and technicalities. Such a finiteness statement would contrast sharply with the situation in dimension $4$.



Homotopy groups of Diff(X) and Homeo(X)
No, the statement about the kernel and cokernel being finite is not true.


For a closed $d$-manifold, $d \neq 4$, smoothing theory identifies the homotopy fibre of
$$B\mathrm{Diff}(M) \longrightarrow B\mathrm{Homeo}(M)$$
with (certain path components of) the space of sections of a bundle
$$Top(d)/O(d) \longrightarrow E \longrightarrow M$$
constructed from te tangent bundle of $M$. Supposing for simplicity that $M$ is parallelisable, this space of sections is equivalent to
$$\mathrm{map}(M, Top(d)/O(d)).$$


So your question is more or less equivalent to asking about the homotopy groups of $Top(d)/O(d)$. Until recently it was not even known whether the homotopy groups of $Top(d)/O(d)$ are finitely-generated, but that was proved by Kupers, in *Some finiteness results for groups of automorphisms of manifolds*. These days quite a bit is known about the rational homotopy groups of these spaces. Using the above formulation of smoothing theory (which goes through unchanged for manifolds with boundary) for the disc $D^d$, you will find most results are stated for
$$B\mathrm{Diff}\_\partial(D^d) \simeq \Omega^d\_0(Top(d)/O(d)).$$
(Here I have used that $B\mathrm{Homeo}\_\partial(D^d) \simeq \*$ by the Alexander trick.) Any result which shows that $B\mathrm{Diff}\_\partial(D^d)$ has a rationally nontrivial homotopy group provides a counterexample to your finiteness question.


For example:


1. $\pi\_i(Top(2n)/O(2n)) \otimes \mathbb{Q}=0$ for $i< 4n-2$, but
$$\pi\_{4n-2}(Top(2n)/O(2n)) \otimes \mathbb{Q}= \mathbb{Q}.$$
(See Kupers--Randal-Williams *On diffeomorphisms of even-dimensional discs*.)
2. $\pi\_i(Top(2n+1)/O(2n+1)) \otimes \mathbb{Q}=0$ for $i< 2n+5$, but
$$\pi\_{2n+5}(Top(2n+1)/O(2n+1)) \otimes \mathbb{Q}= \mathbb{Q}.$$
(This follows from Farrell--Hsiang *On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds*.)


There are some slides from a talk I recently gave at [https://www.dpmms.cam.ac.uk/~or257/MIT2020.pdf](https://www.dpmms.cam.ac.uk/%7Eor257/MIT2020.pdf), which give some more detail about the state of the art.


