Computing All Maps into a Sphere

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 ], i.e., all homotopy classes of such maps.We solve this problem in the stable range, where for some d ≥ 2, we have dimThe 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 ⊂ 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.

Computing simplicial representatives of homotopy group elements

“…Our next goal is to extend the results here to the setting of Čadek et al. (2014a), i.e., to represent, more generally, homotopy classes in [ X , Y ] by explicit simplicial maps from some suitable subdivision to Y (under suitable assumptions that allow us to compute [ X , Y ]) 4…”

“…These results form part of a general effort to understand the computational complexity of topological questions concerning the classification of maps up to homotopy (Čadek et al. 2013a, b, 2014a; Filakovský and Vokřínek 2013) and related questions, such as the embeddability problem for simplicial complexes (a higher-dimensional analogue of graph planarity) (Matoušek et al. 2011, 2014; Čadek et al.…”

“…Apart from the intrinsic importance of homotopy groups, we see this as a step towards the more general goal of computing explicit maps with specific topological properties; instances of this goal include computing explicit representatives of homotopy classes of maps between more general spaces X and Y (a problem raised in Čadek et al. 2014a) as well as computing an explicit embedding of a given simplicial complex into (as opposed to deciding embeddability ). Moreover, these questions are also closely related to quantitative questions in homotopy theory (Gromov 1999) and in the theory of embeddings (Freedman and Krushkal 2014).…”

“…Building on the framework of objects with effective homology by Sergeraert et al, in recent years a variety of new results in computational homotopy theory were obtained (Čadek et al. 2013b, 2014a, b, 2017; Krčál et al. 2013; Vokřínek 2017; Filakovský and Vokřínek 2013; Romero and Sergeraert 2012, 2016), including, in some cases, the first polynomial-time algorithms , by using a refined framework of objects with polynomial-time homology (Krčál et al.…”

“…4Similarly as before, the algorithm in Čadek et al. (2014a) computes [ X , Y ] as the set [ X , P ] for some auxiliary space P (a stage of a Postnikov system for Y ) and represents the elements of as maps from X to P , but not as maps to Y .…”