Ideas of connectivity join with those of shape to describe the topology of the galaxy distribution. This addresses the question much discussed in the 1950s (see Chapter 7) of whether clusters are condensations in an otherwise uniform sea of galaxies or whether clusters are just the edges of voids and underdense regions. The question resurfaced in the 1980s when astronomers noticed fairly large three-dimensional volumes containing relatively few galaxies (Tifft & Gregory, 1976; Kirshner et al., 1981; de Lapparent et al.,1986; Kauffmann & Fairall, 1991). Consequently, much high-energy speculation arose over the origin of voids and cellular structure in the early universe. The main question was: Are clusters or voids the fundamental entities of the galaxy distribution? The answer is: both or neither.

It all depends on how you look at the distribution. If galaxies are the fundamental entities, then clusters and voids are just derivative configurations. If clusters and voids are fundamental, imposed by conditions in the early universe, then galaxies are just derivative markers. If dark matter dominates the universe, the situation becomes even more murky. In any case, topology helps quantify the conditions where relatively underdense or overdense regions dominate. It is most useful, so far, on scales at least several times that of the two-point correlation function (∼ 5 h–1 Mpc where h is the Hubble constant in units of 100 km s–1 Mpc–1).