  There has been considerable recent interest, primarily motivated by problems
in applied algebraic topology, in the homology of random simplicial complexes.
We consider the scenario in which the vertices of the simplices are the points
of a random point process in $\mathbb {R}^d$, and the edges and faces are
determined according to some deterministic rule, typically leading to \v{C}ech
and Vietoris-Rips complexes. In particular, we obtain results about homology,
as measured via the growth of Betti numbers, when the vertices are the points
of a general stationary point process. This significantly extends earlier
results in which the points were either i.i.d. observations or the points of a
Poisson process. In dealing with general point processes, in which the points
exhibit dependence such as attraction or repulsion, we find phenomena
quantitatively different from those observed in the i.i.d. and Poisson cases.
From the point of view of topological data analysis, our results seriously
impact considerations of model (non)robustness for statistical inference. Our
proofs rely on analysis of subgraph and component counts of stationary point
processes, which are of independent interest in stochastic geometry.
