homotopy of sphere maps - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:32:21Z http://mathoverflow.net/feeds/question/50719 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/50719/homotopy-of-sphere-maps homotopy of sphere maps peter franek 2010-12-30T11:48:49Z 2011-01-01T06:08:59Z <p>Hi all. If n>m and I have a continuous map from an n-sphere to an m-sphere, is there a simple way to see whether the map is homotopic to a constant? (For example, is it if and only if the map is not onto? -- this would be probably too simple). For n=m, the answer is "nonzero degre" -- is there a reasanable generalisation of "degree" to different dimensions? Can you recommend me some literature about it? Thanks, Peter</p> http://mathoverflow.net/questions/50719/homotopy-of-sphere-maps/50784#50784 Answer by Dev Sinha for homotopy of sphere maps Dev Sinha 2010-12-31T04:26:40Z 2010-12-31T04:26:40Z <p>The set of maps $S^n \to S^m$ is better known as $\pi_n(S^m)$. The one other (than n=m) relatively easy case is when we tensor by the rational numbers (and thus answer the question "when is f homotopic to g after adding each to itself some number of times?). The answer in this case is they are always homotopic if m is odd or if m is even and $n \neq 2m - 1$. If $m$ is even and $n = 2m-1$ then $f$ and $g$ are homotopic in this weak sense if and only if they both have non-trivial (or both have trivial) classical Hopf invariant.</p> http://mathoverflow.net/questions/50719/homotopy-of-sphere-maps/50828#50828 Answer by Jeff Strom for homotopy of sphere maps Jeff Strom 2010-12-31T19:59:18Z 2011-01-01T06:08:59Z <p>Every map from a sphere to a finite <em>connected</em> CW complex is homotopic to a surjective one. Collapse one half of the sphere to a line segment, use your original map on the sphere created by the collapse, and map the segment surjectively onto the complex (which can be done by the Hahn-Mazurkowicz Theorem).</p>