most recent 30 from http://mathoverflow.net 2013-05-23T19:58:04Z http://mathoverflow.net/feeds/question/52620 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52620/smooth-and-analytic-structures-on-low-dimensional-euclidian-spaces Hugo Chapdelaine 2011-01-20T13:56:49Z 2011-01-20T19:52:40Z

<p>So it is relatively easy to show that there exists only one smooth structure on the real line $\mathbb{R}$. So here are 2 natural questions:</p> <p>Q1: Up to equivalence, is there only one real analytic structure on $\mathbb{R}$? If so, then do we have a simple proof of that? </p> <p>Q2: Where can I find the simplest proofs that there exists only one smooth structure on $\mathbb{R}^2$ and $\mathbb{R}^3$?</p> <p>So I've heard that on $\mathbb{R}^4$ there are infinitly (in fact uncountably) many non-equivalent smooth structures, so what about real analytic strucutres? Is there some kind of moduli space of smooth structures on $\mathbb{R}^4$. if so, in how many ways is it possible to deform a smooth structure into a real analytic one?</p>

http://mathoverflow.net/questions/52620/smooth-and-analytic-structures-on-low-dimensional-euclidian-spaces/52682#52682 Ryan Budney 2011-01-20T19:45:16Z 2011-01-20T19:45:16Z

<p>Regarding Q1, put an analytic Riemann metric on your 1-manifold. Integrating a unit speed vector field gives an analytic diffeomorphism to $\mathbb R$. Another way to prove analytic structures are unique is to notice the same argument that one uses to prove that the group of $C^k$-diffeomorphisms of $\mathbb R$ has the homotopy type of $\mathbb Z_2$ works for analytic diffeomorphisms -- simply take the straight-line homotopy between your original diffeomorphism and either the identity or the negative identity, appropriately. </p> <p>Regarding Q2, I don't know much in the way of really simple proofs. But when $n=2$ you've got the Uniformization Theorem from complex analysis. That's relatively simple. </p>