  This study presents a numerical analysis of the topology of a set of
cosmologically interesting three-dimensional Gaussian random fields in terms of
their Betti numbers $\beta_0$, $\beta_1$ and $\beta_2$. We show that Betti
numbers entail a considerably richer characterization of the topology of the
primordial density field. Of particular interest is that Betti numbers specify
which topological features - islands, cavities or tunnels - define its spatial
structure.
  A principal characteristic of Gaussian fields is that the three Betti numbers
dominate the topology at different density ranges. At extreme density levels,
the topology is dominated by a single class of features. At low levels this is
a \emph{Swiss-cheeselike} topology, dominated by isolated cavities, at high
levels a predominantly \emph{Meatball-like} topology of isolated objects. At
moderate density levels, two Betti number define a more \emph{Sponge-like}
topology. At mean density, the topology even needs three Betti numbers,
quantifying a field consisting of several disconnected complexes, not of one
connected and percolating overdensity.
  A {\it second} important aspect of Betti number statistics is that they are
sensitive to the power spectrum. It reveals a monotonic trend in which at a
moderate density range a lower spectral index corresponds to a considerably
higher (relative) population of cavities and islands.
  We also assess the level of complementary information that Betti numbers
represent, in addition to conventional measures such as Minkowski functionals.
To this end, we include an extensive description of the Gaussian Kinematic
Formula (GKF), which represents a major theoretical underpinning for this
discussion.
