Computing all maps into a sphere

TL;DR

This paper presents an algorithm for computing all homotopy classes of continuous maps from finite simplicial complexes into spheres within the stable range, combining classical algebraic topology tools with effective computational methods.

Contribution

It introduces a computational approach to determine all homotopy classes of maps into spheres in the stable range, integrating classical and algorithmic topology techniques.

Findings

Algorithm computes [X,Y] in polynomial time for fixed d

Extends to problems like map extension and Z_2-index in the stable range

Outside the stable range, the extension problem is undecidable

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 <= 2d - 2 and Y is (d - 1)-connected; in particular, Y can be the d-dimensional sphere S^d. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X = S^1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A of X and a map A -> 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.