Quarter Turn Compositions and PETs

This chapter proves a general compactification theorem for quarter turn compositions. It is organized as follows. Section 15.2 proves a well-known result from linear algebra which will help with the material in the following section. Section 15.3 defines the map Ψ‎: S → Ŝ and study the dimension of its image as a function of the parameters of Τ‎. Recall that Τ‎ is a composition of shears and quarter turn maps. Section 15.4 establishes Lemma 15.6, which shows that Ψ‎ interacts in the desired way with shears. Ψ‎15.5 establishes Lemma 15.7, which does the same thing for quarter turn maps. Ψ‎15.6 combines Lemmas 15.6 and 15.7 to prove Theorem 15.1.