Chapter 0 Prequel This Prequel sets forth – with references, but with few proofs – important background results covered by neither Rourke and Sanderson nor Munkres. Readers may want to briefly familiarize themselves with the contents of this chapter and then begin their serious study with Chapter 1. Chapter 0 can be used as a reference for topics that arise later and consulted as needed. The prerequisites covered in this chapter should be enough to carry the reader through the first five chapters of the book. Beyond that point, additional deep material occasionally will be interwoven, without proof, to present a complete picture of current developments. 0.1. More definitions and notation The n-cube In is the n-fold product [−1, 1]n. Following Rourke and Sander- son (1972, page 4), we consistently use I1 to denote the interval [−1, 1], but sometimes use I to denote the interval [0, 1]. Whether I denotes [0, 1] or [−1, 1] should be clear from the context. Of course In is homeomorphic to Bn, so the n-cube is an n-cell. In some contexts a k-cell will also be called a k-disk and will be denoted Dk. Let X and Y be two spaces with base points x0 and y0, respectively. The wedge (or wedge sum) of X and Y is the quotient of the disjoint union X Y obtained by identifying x0 and y0. The wedge of X and Y is denoted X ∨ Y and might also be called the one-point union of X and Y the wedge of a finite number of circles is often called a bouquet of circles (Figure 0.1). Upper half space R+ n consists of all the points in Rn whose last coordinate is nonnegative i.e., Rn + = {x1,x2,...,xn| each xi is a real number and xn ≥ 0}. 1 http://dx.doi.org/10.1090/gsm/106/01

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