A Random Bockstein Operator

TL;DR

This paper explores the probabilistic properties of the Bockstein operator within algebraic topology, addressing a gap in understanding how cohomology operations behave in high-dimensional data analysis.

Contribution

It introduces a novel study of the Bockstein operator's probabilistic properties, expanding the theoretical framework of topological data analysis tools.

Findings

Analyzes the algebraic behavior of the Bockstein operator under randomness

Provides foundational insights into cohomology operations in probabilistic settings

Highlights the need for further research on cohomology-based methods in data analysis

Abstract

As more of topology's tools become popular in analyzing high dimensional data sets, the goal of understanding the underlying probabilistic properties of these tools becomes even more important. While much attention has been given to understanding the probabilistic properties of methods that use homological groups in topological data analysis, the probabilistic properties of methods that employ cohomology operations remain unstudied. In this paper, we investigate the Bockstein operator with randomness in a strictly algebraic setting.