  We compute the homology of random \v{C}ech complexes over a homogeneous
Poisson process on the d-dimensional torus, and show that there are, coarsely,
two phase transitions. The first transition is analogous to the
Erd\H{o}s-R\'enyi phase transition, where the \v{C}ech complex becomes
connected. The second transition is where all the other homology groups are
computed correctly (almost simultaneously). Our calculations also suggest a
finer measurement of scales, where there is a further refinement to this
picture and separation between different homology groups.
