Finding Bounded Simplicial Sets with Finite Homology

TL;DR

This paper introduces a method to find smaller, bounded simplicial sets with finite homology that are equivalent in a p-local sense, improving the efficiency of homotopy group computations in algebraic topology.

Contribution

The paper presents a procedure to produce bounded simplicial sets with finite homology, reducing the dependence of homotopy group algorithms on input size.

Findings

Bounded simplicial sets can be constructed with size depending only on dimension and homology.

The method enables more efficient p-local homology computations.

Preprocessing isolates size dependence, improving overall algorithm performance.

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.