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))
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