Homological connectivity in random Čech complexes

We study the homology of random Čech 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 Erdős-Rényi graph and random geometric graphs. We first prove that there is a sequence of sharp phase transitions describing homological connectivity in different dimensions. Next, we analyze the behavior of the complex inside each of the critical windows. 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.