Smooth structures on PL 4-manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:53:53Z http://mathoverflow.net/feeds/question/7892 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7892/smooth-structures-on-pl-4-manifolds Smooth structures on PL 4-manifolds John Francis 2009-12-05T20:16:47Z 2009-12-05T20:35:14Z <p>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 4-manifold is contractible. (I think it's known that this map is at least 4-connected, which shows that the space of smooth structures on any PL 4-manifold is nonempty and connected.)</p> http://mathoverflow.net/questions/7892/smooth-structures-on-pl-4-manifolds/7894#7894 Answer by Ryan Budney for Smooth structures on PL 4-manifolds Ryan Budney 2009-12-05T20:26:03Z 2009-12-05T20:35:14Z <p>Very little is known about that question, the same smoothing theory gives something that I'm trying to get people to call "The Cerf-Morlet Comparison Theorem"</p> <p>$$Diff(D^n) \simeq \Omega^{n+1}(PL(n)/O(n))$$</p> <p>$Diff(D^n)$ 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 path-connected or not. Very little is known about the homotopy-type of $Diff(D^4)$, no seriously informative statements other than that homotopy-equivalence. I wrote up a paper where I described in detail the iterated loop-space structure and how it arrises naturally. Moreover, I described how that iterated loop-space 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. </p> <p>There are several natural connections here, one of the big ones being that $Diff(D^n)$ has the homotopy-type of the space of round metrics on $S^n$ -- ie the subspace of the affine-space of Riemann metrics on $S^n$, the subspace is specified by the condition that "$S^n$ with this metric is isometric to the standard $S^n$." </p>