Introduction: The Main Problem xvii
the subject is introduced and carefully examined in Chapter 3. The re-
maining chapters strive to systematically investigate the central embedding
problems. That investigation is organized by codimension. The codimen-
sion of an embedding e : X Y is defined by codim(e) = dim Y dim X,
the difference between the ambient dimension and the dimension of the em-
bedded space. Generally speaking, the greater the codimension the easier it
is to prove positive theorems about embeddings. Chapter 4 treats the trivial
range, the range in which the codimension of the embedded space exceeds its
dimension, where the most general theorems hold. Next, Chapter 5 moves
on to codimension three, to which many trivial-range theorems extend with
appropriate modifications. However, very few of the codimension-three the-
orems extend to codimension two, so Chapter 6 is largely devoted to the
construction of codimension-two counterexamples. In codimension one the
situation changes once more, and again there are many positive results,
which form the subject of Chapter 7. The book concludes in Chapter 8 with
a quick description of some codimension-zero results.
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