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This study presents a numerical analysis of the topology of a set ofcosmologically interesting three-dimensional Gaussian random fields in terms oftheir Betti numbers $\beta_0$, $\beta_1$ and $\beta_2$. We show that Bettinumbers entail a considerably richer characterization of the topology of theprimordial density field. Of particular interest is that Betti numbers specifywhich topological features - islands, cavities or tunnels - define its spatialstructure.  A principal characteristic of Gaussian fields is that the three Betti numbersdominate the topology at different density ranges. At extreme density levels,the topology is dominated by a single class of features. At low levels this isa \emph{Swiss-cheeselike} topology, dominated by isolated cavities, at highlevels a predominantly \emph{Meatball-like} topology of isolated objects. Atmoderate 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 oneconnected and percolating overdensity.  A {\it second} important aspect of Betti number statistics is that they aresensitive to the power spectrum. It reveals a monotonic trend in which at amoderate density range a lower spectral index corresponds to a considerablyhigher (relative) population of cavities and islands.  We also assess the level of complementary information that Betti numbersrepresent, in addition to conventional measures such as Minkowski functionals.To this end, we include an extensive description of the Gaussian KinematicFormula (GKF), which represents a major theoretical underpinning for thisdiscussion.

Topology and Geometry of Gaussian random fields I: on Betti Numbers, Euler characteristic and Minkowski functionals