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?
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