Quantifying Homology Classes

TL;DR

This paper introduces a method to quantify and optimize the measurement of homology classes in topological spaces, providing algorithms and complexity analysis for basis selection and localization.

Contribution

It defines a new size measure for homology classes, proposes a greedy algorithm for optimal basis computation, and explores localization methods with hardness results.

Findings

Developed a size measure for homology classes using relative homology.

Provided a greedy algorithm for computing an optimal basis with complexity analysis.

Discussed localization techniques and proved related hardness results.

Abstract

We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in $O(\beta^4 n^3 \log^2 n)$ time, where $n$ is the size of the simplicial complex and $\beta$ is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results.