The Plaid Master Picture Theorem

Abstract

This chapter aims to prove Theorem 1.4 and Theorem 0.3, the Plaid Master Picture Theorem. Both of these results are deduced from Theorem 8.2, which says that the union PLA of plaid polygons is generated by an explicitly defined tiling classifying space (ලA, XP). Moreover, there is a nice space X which has the individual spaces XP as rational slices. The space X has a partition into convex polytopes, and one obtains the partition of XP by intersecting the relevant slice with this partition. Section 8.2 describes the space X. Section 8.3 describes the partition of X into convex integer polytopes. The partition is called the checkerboard partition. Section 8.4 explains the classifying map Φ‎A : Π‎ → XP. Section 8.5 states Theorem 8.2 and deduces Theorem 1.4 and Theorem 0.3 from it.