**3.4. Genus**

A promising way of classifying the structure of the universe is to see how the topology of the surfaces of constant density in the universe varies with the value of the density (Gott, Melott and Dickinson, 1986; Weinberg, Gott and Melott, 1987). The topological genus of the set of points enclosed by surfaces of constant density is defined as the number of holes minus the number of isolated regions (though this is not how it is calculated). Thus a toroidal region has a genus of -1 and an isolated sphere has a genus of 0. The genus signature of the distribution is calculated as a function of the density threshold at which the surfaces are defined. Several authors, starting with Doroshkevich (1970) calculated the genus signature for a windowed Gaussian random density field as a function of the threshold defining the surfaces:

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is measured in terms of the variance of the field seen through the selected window. The constant of proportionality depends on the power spectrum of the windowed density distribution. If the distribution is non-Gaussian, this would show up in the genus signature.

This has been discussed at length in two recent papers Gott et al. (1989) and Melott et al. (1989). The main technical problem is that of relating the observed two dimensional sections of three dimensional data to the three dimensional situation. In the data paper (Gott et al, 1989), catalogues of all kinds of objects ranging from dwarf galaxies to Abell clusters are used. It is found that the universe is "sponge-like" on largest scales, with no evidence for a bubble-like structure which, it is claimed, could have been detected if present. There are voids, but they are interconnected, as opposed to being surrounded on all sides by galaxies. On smaller scales, the topology has a tendency to become "meatball-like".

Moore (1991)
has looked at the QDOT redshift survey, smoothed over a
variety of windows. The genus signature is clear on scales of
20*h*^{-1}
Mpc. and 40*h*^{-1} Mpc. and is consistent with a Gaussian
distribution of density fluctuations taken from a power law power
spectrum with spectral index
*n* = - 0.79 ± 0.35 on those scales. This appears to be a
variance with the general hypothesis that the spectrum has the
Harrison-Zel'dovich form on such scales, and in particular with the
standard CDM models.