The Shape of M
Suppose p is a parameter point on a conflict line. We translate the sociological notion of „conflicting opinions at p“ into the mathematical notion of bimodality of Fp; in other words, we assume Fp has two local maxima, and so M is two-sheeted over a neighborhood of p. Now globalize this notion by assuming that F is bimodal over neighborhoods N1, N2 of the two conflict lines L1, L2 , and is unimodal otherwise. We call N1, N2 the conflict regions; they are shown shaded in Figure 20.3. As we leave either side of a conflict region one of the two modes is preserved and the other has to disappear; we assume, further, that one of the modes is preserved on one side and the other on the other side. Therefore S has an S-shaped fold over N1. Meanwhile L2 terminates in the point K, and by Hypothesis 3 the only way an S-shaped fold can terminate is in a cusp (Markus 1977; Zeeman n.d.); therefore, S has a cusp singularity at K. Summarizing our assumptions and deductions into a single hypothesis we obtain Hypothesis 4.
The projection x: S → P has fold curves and a cusp singularity as shown in Figure 20.3. The reader will observe that we have drawn S in Figure 20.3 as if it were sitting in three dimensions rather than in the (2 + n)-dimensional space P x X. We justify this in a later section. Meanwhile notice that our discussion of folds and cusps is independent of the ambient space in which S happens to be sitting. The following deductions from Hypothesis 4 are also independent of the ambient space. M has two components M1 and M2, whose projections overlap on N1. The projection of M2 overlaps itself on N2. The complementary subsurface S-M has two components S1, S2 shown shaded in Figure 20.3, that project onto N1, N2; however, the components S1, S2 are sociologically meaningless because they represent saddle points of F, around which no opinion points cluster.