Covering maps on Euclidean spaces and spheres - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:53:22Z http://mathoverflow.net/feeds/question/18379 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18379/covering-maps-on-euclidean-spaces-and-spheres Covering maps on Euclidean spaces and spheres Julgyz Harzum 2010-03-16T14:13:40Z 2010-03-16T17:17:20Z <p>Hello. I have two questions.</p> <ol> <li><p>Does there exist an exactly 2-fold covering map $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ ?</p></li> <li><p>Does there exist an exactly 2-fold covering map $g:S^{n}\rightarrow S^{n}$ ?</p></li> </ol> <p>Here $S^{n}$ is the unit $n$-sphere, $S^{n}={x\in\mathbb{R}^{n+1}: \|x\|=1}$.</p> <p>Great thanks.</p> http://mathoverflow.net/questions/18379/covering-maps-on-euclidean-spaces-and-spheres/18401#18401 Answer by Anton Geraschenko for Covering maps on Euclidean spaces and spheres Anton Geraschenko 2010-03-16T17:17:20Z 2010-03-16T17:17:20Z <p>I think Pete should have made his comment an answer, so I'll do it for him.</p> <p>Theorem 1.38 of Hatcher's <a href="http://www.math.cornell.edu/~hatcher/AT/ATpage.html" rel="nofollow">Topology</a> says that connected coverings of a (locally path-connected, and semilocally simply-connected) topological space $X$ are in bijection with conjugacy classes of subgroups of $\pi_1(X)$.</p> <p>Since $\pi_1(X)$ is trivial for $X=\mathbb{R}$ or $X=S^n$ ($n>1$), there are no connected coverings.</p>