Computing simplicial representatives of homotopy group elements

Abstract

A central problem of algebraic topology is to understand the homotopy groupsd(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 group1(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., with1(X) trivial), compute the higher homotopy group π (X) for any given d ≥ 2. However, these algorithms come with a caveat: They compute the isomorphism type of πd(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 πd(X). Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere Sd 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 πd(X) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere Sd 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 πd(X), the size of the triangulation of Sd on which the map is defined is exponential in size(X).