Introduction: The

Main Problem

Prerequisites. What background is needed for reading this text? Chiefly,

a knowledge of piecewise linear topology. For many years the standard

reference in that area has been the text Introduction to Piecewise-Linear

Topology, by C. P. Rourke and B. J. Sanderson (1972), and we assume

familiarity with much of their book. To be honest, that book presumes

extensive understanding of both general and algebraic topology; as a con-

sequence we implicitly are assuming those subjects as well. In an attempt

to limit our presumptions, we specifically shall take as granted the results

from two fairly standard texts on general and algebraic topology, both by

J. R. Munkres—namely, his Topology: Second Edition (2000) and Elements

of Algebraic Topology (1984), each of which can be treated quite effectively

in a year-long graduate course.

Unfortunately, even those three texts turn out to be insuﬃcient for all

our needs. The purpose of the initial Chapter 0, the Prequel, is to correct

that deficiency.

Basic Terminology. The notation laid out in this subsection should be

familiar to those who have read Rourke and Sanderson’s text. Neverthe-

less, we spell out the essentials needed to fully understand the forthcoming

discussion of the primary issues addressed in this book.

Here R denotes the set of real numbers and

Rn

denotes n-dimensional

Euclidean space, the Cartesian product of n copies of R. For 1 ≤ k n we

regard

Rk

as included in

Rn

in the obvious way, as the subset containing all

points whose final (n − k)-coordinates are all equal to zero.

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