xvi Introduction: The Main Problem

The problems listed above are the main ones that will occupy attention

in this text. They all can be viewed as uniqueness questions in the sense that

they ask whether given embeddings are equivalent. There are also existence

questions for embeddings, which will be studied alongside the uniqueness

questions. We identify two such: one global, the other local.

Existence Problem. Given a map f : X → Y, is f homotopic to a topo-

logical embedding or a PL embedding?

Approximation Problem. Which topological embeddings of X in Y can

be approximated by PL embeddings?

The flatness concept has a local version. A topological embedding e :

M → N of a k-dimensional manifold M into an n-dimensional manifold N

is locally flat at x ∈ M if there exists a neighborhood U of e(x) in N such

that (U, U ∩ e(M))

∼

=

(Rn, Rk).

An embedding is said to be locally flat if

it is locally flat at each point x of its domain. The last two problems have

local variations: for example, one can ask whether a map of manifolds is

homotopic to a locally flat embedding or whether a topological embedding

of manifolds can be approximated by locally flat embeddings.

When considering an embedding e : X → Y , the dimension of Y is called

the ambient dimension. Almost all of the examples and theorems in this

book involve embeddings in manifolds of ambient dimension three or more.

We skip dimension two because classical results like the famous Sch¨onflies

theorem (Theorem 0.11.1) imply that no nonstandard local phenomena arise

in conjunction with embeddings into manifolds of that dimension.

While isolated examples of wild embeddings were discovered earlier, the

work of R. H. Bing in the 1950s and 1960s revealed the pervasiveness of

wildness in dimensions three and higher. His pioneering work led to a pro-

liferation of embedding results, first concentrating on dimension three, but

soon expanding to include higher dimensions as well. The subject of topo-

logical embeddings is now a mature branch of geometric topology, and this

book is meant to be a summary and exposition of the fundamental results

in the area.

Organization. As mentioned earlier, the initial Chapter 0 addresses back-

ground matters. The real beginning, Chapter 1, treats knottedness, tame-

ness and local flatness; it provides examples of knotted, PL codimension-two

sphere pairs in all suﬃciently large dimensions, and it delves into the local

homotopy properties of nicely embedded objects. Chapter 2 presents the

basic examples that motivate the study and offers context for theorems to

come later; it also includes several flatness theorems that can be proved

without the use of engulfing. Engulfing – the fundamental technical tool for