Is it known whether $O(4) \to PL(4)$, the map from the orthogonal group to the group of piecewise linear homeomorphisms of $\mathbb{R}^4$, is a homotopy equivalence? By smoothing theory for PL manifolds, this is equivalent to whether the space of smooth structures on a PL 4manifold is contractible. (I think it's known that this map is at least 4connected, which shows that the space of smooth structures on any PL 4manifold is nonempty and connected.)
Very little is known about that question, the same smoothing theory gives something that I'm trying to get people to call "The CerfMorlet Comparison Theorem"
$$ Diff(D^n) \simeq \Omega^{n+1}(PL(n)/O(n)) $$
$Diff(D^n)$ is the group of diffeomorphisms of the $n$ball where the diffeomorphisms are pointwise fixed on the boundary. Nobody knows if $Diff(D^4)$ is pathconnected or not. Very little is known about the homotopytype of $Diff(D^4)$, no seriously informative statements other than that homotopyequivalence. I wrote up a paper where I described in detail the iterated loopspace structure and how it arrises naturally. Moreover, I described how that iterated loopspace structure relates to various natural maps. That's my main relation to to topic. The paper is called "Little cubes and long knots" and is on the arXiv. I elaborate on some of these issues in the paper "A family of embedding spaces", also on the arXiv.
There are several natural connections here, one of the big ones being that $Diff(D^n)$ has the homotopytype of the space of round metrics on $S^n$  ie the subspace of the affinespace of Riemann metrics on $S^n$, the subspace is specified by the condition that "$S^n$ with this metric is isometric to the standard $S^n$."

$\begingroup$ Ryan, I always thought that path components of $Diff(D^4,\partial D^4)$ are the same as the group of twist 5spheres, which is a single element. ? $\endgroup$ – Andrey Gogolev Dec 6 '09 at 1:26

$\begingroup$ The group of twisted $n$spheres is isomorphic to the quotient of $\pi_0 Diff(S^{n−1})$ modulo the subgroup of diffeomorphisms that extend over $D^n$  saying they extend over $D^n$ is saying they are pseudoisotopic to the identity. Cerf's pseudoisotopy theorem says pseudoisotopy is equivalent to isotopy in dimensions $5$ and larger. ie: $\pi_0 Diff(D^n)$ is only known to be the same as the group of exotic $n+1$ spheres if $n \geq 5$ $\endgroup$ – Ryan Budney Dec 6 '09 at 7:45

$\begingroup$ I see. Thank you! Are there any "concordance implies isotopy" theorems in dimension 4? $\endgroup$ – Andrey Gogolev Dec 6 '09 at 16:29

$\begingroup$ Nothing is jumping to mind at the moment. At least, if you want to keep everything smooth. Most concordanceisotopy problems in dimension 4 that I'm thinking of are all big open problems. $\endgroup$ – Ryan Budney Dec 11 '09 at 1:14

$\begingroup$ A paper just went up on the arXiv claiming some very strong results about the diffeomorphism group. In particular, they show that $\text{Diff}(D^4, \partial)$ is not contractible, though can say nothing about path components. (I have not read it yet and can make no evaluation of the proof.) $\endgroup$ – Mike Miller Dec 7 '18 at 2:42