Computing simplicial representatives of homotopy group elements

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).