Proof of the Main Result

This chapter puts together the ingredients from the last three chapters—the Segment Lemma, the Horizontal Lemma, and the Vertical Lemma—and proves Theorem 8.2. The three technical lemmas do not mention the partition of the space X at all, but they do give a lot of control over how the nature of the particles tracked by points in the plaid grid Π‎ influences the image of such grid points under the classifying map Φ‎. What remains is to compare the three results above to the partition and determine whether everything matches. The remainder of the chapter is organized as follows. Section 12.2 carries out the program to show that these containers are each a union of two prisms. Section 12.3 discusses some extra symmetry of the partition. Section 12.5 compares the prism containers in the vertical case to the partition of X and deduces that Theorem 8.2 is true for the vertical unit integer segments. Section 12.4 compares the prism containers in the vertical case to the partition of X and deduces that Theorem 8.2 is true for the horizontal unit integer segments. The two results together complete the proof.