Abstract
A central problem of algebraic topology is to understand the homotopy groupspi _d(X) of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental grouppi _1(X) of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with pi _1(X) trivial), compute the higher homotopy group pi _d(X) for any given dge 2. However, these algorithms come with a caveat: They compute the isomorphism type of pi _d(X), dge 2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of pi _d(X). Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere S^d to X has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected space X, computes pi _d(X) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere S^d to X. For fixed d, the algorithm runs in time exponential in mathrm {size}(X), the number of simplices of X. Moreover, we prove that this is optimal: For every fixed dge 2, we construct a family of simply connected spaces X such that for any simplicial map representing a generator of pi _d(X), the size of the triangulation of S^d on which the map is defined, is exponential in mathrm {size}(X).
Highlights
One of the central concepts in topology are the homotopy groups πd (X ) of a topological space X
The algorithm runs in time exponential in size(X ), the number of simplices of X. We prove that this is optimal: For every fixed d ≥ 2, we construct a family of connected spaces X such that for any simplicial map representing a generator of πd (X ), the size of the triangulation of Sd on which the map is defined, is exponential in size(X )
A fundamental computational result about homotopy groups is negative: There is no algorithm to decide whether the fundamental group π1(X ) of a finite simplicial complex X is trivial, i.e., whether every continuous map from the circle S1 to X can be continuously contracted to a point; this holds even if X is restricted to be 2-dimensional
Summary
One of the central concepts in topology are the homotopy groups πd (X ) of a topological space X. On the negative side, computing πd (X ) is #P-hard if d is part of the input (Anick 1989; Cadek et al 2013b) (and, W[1]-hard with respect to the parameter d Matoušek 2014), even if X is restricted to be 4-dimensional These results form part of a general effort to understand the computational complexity of topological questions concerning the classification of maps up to homotopy (Cadek et al 2013a, b, 2014a; Filakovský and Vokrínek 2013) and related questions, such as the embeddability problem for simplicial complexes (a higher-dimensional analogue of graph planarity) (Matoušek et al 2011, 2014; Cadek et al 2017)
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