Hi all. If n>m and I have a continuous map from an nsphere to an msphere, 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

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 $n = 2m1$ then $f$ and $g$ are homotopic in this weak sense if and only if they both have nontrivial (or both have trivial) classical Hopf invariant. 


Every map from a sphere to a finite connected 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 HahnMazurkowicz Theorem). 

