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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