Computing Simplicial Representatives of Homotopy Group Elements

Abstract

A central problem of algebraic topology is to understand the homotopy groups π 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 group π 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 π 1 (X) trivial), compute the higher homotopy groupHowever, these algorithms come with a caveat: They compute the isomorphism type of π d (X), d ≥ 2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of π d (X). Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere S d to X has been an important open question in the emerging field of computational homotopy theory.Here we present an algorithm that, given a simply connected simplicial complex X, computes π 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 size(X), the number of simplices of X. Moreover, we prove that this is optimal: For every fixed d ≥ 2, we construct a family of simply connected simplicial complexes X such that for any simplicial map representing a generator of π d (X), the size of the triangulation of S d on which the map is defined is exponential in size(X).