Year: 2011
Type: preprint
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.
Source: arXiv (Cornell University)
Institutions: Institut Fourier, Institute of Science and Technology Austria
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Minors and Embeddability of Simplicial Complexes, G3193747090, Center of excellence - Institute for theoretical computer science (CE-ITI), Combinatorial and Computational Aspects of Embeddings, Embeddings in Higher Dimensions: Algorithms and Combinatorics, Grundlagenforschung zur computerbasierten Psychodiagnostik., G4969897304, G5599649393, DISCRETE AND CONVEX GEOMETRY: CHALLENGES, METHODS, APPLICATIONS, G580951548, G628552264, G993971353
OpenAlex ID: W4299945953