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.,

R+

n

= {x1,x2,...,xn | each xi is a real number and xn ≥ 0}.

1

http://dx.doi.org/10.1090/gsm/106/01