Computing all maps into a sphere

Abstract

Given topological spaces X and Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X -> Y . We consider a computational version, where X, Y are given as finite simplicial complexes, and the goal is to compute [X,Y], i.e., all homotopy classes of such maps. We solve this problem in the stable range, where for some d >= 2, we have dim X Y and ask whether it extends to a map X -> Y, or computing the Z_2-index---everything in the stable range. Outside the stable range, the extension problem is undecidable.