xiv Introduction: The Main Problem
We use
Bn
to denote the standard n-ball (or n-cell) in
Rn,
Int
Bn
to
denote its interior, and
Sn−1
to denote the standard (n − 1)-sphere, the
boundary,
∂Bn,
of
Bn.
Specifically,
Bn
= {x1,x2,...,xn ∈
Rn
| x1
2
+ x2
2
+ · · · + xn
2
≤ 1},
Int
Bn
= {x1,x2,...,xn ∈
Rn
| x1
2
+ x2
2
+ · · · + xn
2
1}, and
Sn−1
=
∂Bn
= {x1,x2,...,xn ∈
Rn
| x1
2
+ x2
2
+ · · · + xn
2
= 1}.
We call any space homeomorphic to
Bn
or
Sn−1
an n-cell or an (n − 1)-
sphere, respectively. The k-ball
Bk
is defined as a subset of
Rk,
but for
each k n the inclusion
Rk
⊂
Rn
determines a standard k-ball
Bk
and a
standard (k − 1)-sphere
Sk−1
in
Rn
as well.
All simplicial complexes and CW complexes are assumed to be locally
finite. A polyhedron is the underlying space of a simplicial complex. While
a simplicial complex K and the underlying polyhedron |K| are two differ-
ent things, we will not always maintain this distinction in our terminology.
Piecewise linear is abbreviated PL.
An n-dimensional (topological) manifold is a separable metric space in
which each point has a neighborhood that is homeomorphic to
Rn.
Such a
neighborhood is called a coordinate neighborhood of the point.
The Main Problem. The central topic in this text is topological embed-
dings. Formally, an embedding of one topological space X in another space
Y is nothing more than a homeomorphism of X onto a subspace of Y . The
domain X is called the embedded space and the target Y is called the am-
bient space. Two embeddings λ,λ : X → Y are equivalent if there exists a
(topological) homeomorphism Θ of Y onto itself such that Θ ◦ λ = λ . The
main problem in the study of topological embeddings is:
Main Problem. Which embeddings of X in Y are equivalent?
In extremely rare circumstances all pairs of embeddings are equivalent. For
instance, if X is just a point, the equivalence question for an arbitrary
pair of embeddings of X in a given space Y amounts to the question of
homogeneity of Y , which has an affirmative answer whenever, for example,
Y is a connected manifold.
Ordinarily, then, our interest will turn to conditions under which embed-
dings are equivalent, and we will limit attention to reasonably well-behaved
spaces X and Y . Specifically, in this book the embedded space X will ordi-
narily be a compact
polyhedron1
and the ambient space Y will always be a
manifold, usually a piecewise linear (abbreviated PL) manifold. If there are
embeddings of the polyhedron X in the PL manifold Y that are homotopic
1A
major exception is the study of embeddings of the Cantor set.
