Introduction: The Main Problem xv

but not equivalent, then X is said to knot in Y . For given polyhedra X and

Y , it is often possible to identify a distinguished class of PL embeddings of

X in Y that are considered to be unknotted; any PL embedding that is not

equivalent to an unknotted embedding is then said to be knotted.

While we do place limitations on the spaces considered, we intentionally

include the most general kinds of topological embeddings in the discussion.

Let X be a polyhedron and let Y be a PL manifold. An embedding X → Y

is said to be a tame embedding if it is equivalent to a PL embedding; the

others are called wild. For embeddings of polyhedra the Main Problem splits

off two fundamental special cases, one called the Taming Problem and the

other the (PL) Unknotting Problem.

Taming Problem. Which topological embeddings of X in Y are equivalent

to PL embeddings?

Unknotting Problem. Which PL embeddings of X in Y are equivalent?

The point is, for tame embeddings the Main Problem reduces to the

Unknotting Problem, and PL methods provide effective – occasionally com-

plete – answers to the latter. As we shall see, local homotopy properties

give very precise answers to the Taming Problem. This also means that

local homotopy properties make detection of wildness quite easy. There are

related crude measures that adequately differentiate certain types of wild-

ness, but the category of wild embeddings is highly chaotic. In fact, at the

time of this writing very little effort had been devoted to classifying in any

systematic way the wild embeddings of polyhedra in manifolds.

A closed subset X of a PL manifold N is said to be tame (or, tame as

a subspace) if there exists a homeomorphism h of N onto itself such that

h(X) is a subpolyhedron; X itself is wild if it is homeomorphic to a simplicial

complex but is not tame. Here the focus is more on the subspace X than

on a particular embedding. One can provide a direct connection, of course:

a closed subset X of a PL manifold N is tame as a subspace if and only if

there exist a polyhedron K and a homeomorphism g : K → X such that

λ = inclusion ◦ g : K → N is a tame embedding.

We say that a k-cell or (k − 1)-sphere X in

Rn

is flat if there exists

a homeomorphism h of

Rn

such that h(X) is the standard object of its

type. Generally, whenever we have some standard object S ⊂

Rn

and a

subset X of

Rn

homeomorphic to S, we will say that X is flat if there is

a homeomorphism h :

Rn

→

Rn

such that h(X) = S. In other words, S

represents the preferred copy in

Rn,

and another copy in

Rn

is flat if it is

ambiently equivalent (setwise) to S.

Flatness Problem. Under what conditions is a cell or sphere in

Rn

flat?