Abstract
Given topological spaces X, 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], that is, all homotopy classes of such maps.
We solve this problem in the stable range, where for some d > 1, we have dim X Y and ask whether it extends to a map X -> Y, or computing the Zeta(2)-index-everything in the stable range. Outside the stable range, the extension problem is undecidable.