Computing simplicial representatives of homotopy group elements

A central problem of algebraic topology is to understand the homotopy groups 𝜋𝑑(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 𝜋1(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 𝜋1(𝑋) trivial), compute the higher homotopy group 𝜋𝑑(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 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 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 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) .