Computing simplicial representatives of homotopy group elements

A central problem of algebraic topology is to understand the homotopy groups[Formula: see text] 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[Formula: see text] 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 [Formula: see text] trivial), compute the higher homotopy group [Formula: see text] for any given [Formula: see text] . However, these algorithms come with a caveat: They compute the isomorphism type of [Formula: see text] , [Formula: see text] as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of [Formula: see text] . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere [Formula: see text] 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 [Formula: see text] and represents its elements as simplicial maps from a suitable triangulation of the d-sphere [Formula: see text] to X. For fixed d, the algorithm runs in time exponential in [Formula: see text] , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed [Formula: see text] , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of [Formula: see text] , the size of the triangulation of [Formula: see text] on which the map is defined, is exponential in [Formula: see text] .