Finding Bounded Simplicial Sets with Finite Homology

Abstract: The central problem in computational algebraic topology is the computation of the homotopy groups of a given space, represented as a simplicial set. Algorithms have been found which achieve this, but the running times depend on the size of the input simplicial set. In order to reduce this dependence on the simplicial set chosen, we describe in this paper a procedure which, given a prime $p$ and a finite, simply-connected simplicial set with finite integral homology, finds a $p$-locally equivalent simplicial set with size upper bounded by a function of dimension and homology. Using this in conjunction with the above algorithm, the $p$-local homology can be calculated such that the running time dependence on the size of the initial simplicial set is contained in a separate preprocessing step.