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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