Let as have a continuous map from a topological space X to an n-sphere, realized as a simplicial map between simplicial complexes (for example), i.e. represented by a finite set of data. Is there an algorithm for determining, whether or not the map is homotopic to a constant map?
I know that for maps from a circle to a 2-complex, this is impossible (due to the word problem). But is it possible for maps "to the sphere"? If the dimension of X is less then 2n-2, a group structure may be defined on the cohomotopy space $\pi^n(X)$ and this group is always commutative, so, the obstacle may be not so hard as in case of a "fundamental group" element.
Sorry if this is not too exact, I hope it makes at least some sense :-).
Peter