  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.
