# Homological Connectivity in Random \v{C}ech Complexes

## TL;DR

This paper investigates the phase transition and critical phenomena in the homology of random Čech complexes generated by Poisson processes, revealing sharp thresholds, unique cycle shapes, and Poisson convergence of obstructions.

## Contribution

It provides a comprehensive high-dimensional analysis of homological connectivity, including phase transition characterization, critical window behavior, and classification of critical points using Morse theory.

## Key findings

- Identified a sharp phase transition for homological connectivity.
- Characterized the shape of cycles obstructing connectivity.
- Proved convergence of obstruction counts to a Poisson process.

## Abstract

We study the homology of random \v{C}ech complexes generated by a homogeneous Poisson process. We focus on 'homological connectivity' - the stage where the random complex is dense enough, so that its homology "stabilizes" and becomes isomorphic to that of the underlying topological space. Our results form a comprehensive high-dimensional analogue of well-known phenomena related to connectivity in the Erd\H{o}s-R\'enyi graph and random geometric graphs. We first prove that there is a sharp phase transition describing homological connectivity. Next, we analyze the behavior of the complex in the critical window. We show that the cycles obstructing homological connectivity have a very unique and simple shape. In addition, we prove that the process counting the last obstructions converges to a Poisson process. We make a heavy use of Morse theory, and its adaptation to distance functions. In particular, our results classify the critical points of random distance functions according to their exact effect on homology.