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 $S^n$." 

