  In this paper we compute the cohomology groups of the second Voronoi
compactification of the moduli space of abelian fourfolds in all degrees with
the exception of the middle degree 10. We also compute the cohomology groups of
the perfect cone compactification in degree < 10. The main tool is the
investigation of the strata of the compactification corresponding to
semi-abelic varieties with constant torus rank.
