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.