xiv Introduction: The Main Problem We use Bn to denote the standard n-ball (or n-cell) in Rn, Int Bn to denote its interior, and Sn−1 to denote the standard (n − 1)-sphere, the boundary, ∂Bn, of Bn. Specifically, Bn = {x1,x2,...,xn∈ Rn | x1 2 + x2 2 + · · · + xn 2 ≤ 1}, Int Bn = {x1,x2,...,xn∈ Rn | x2 1 + x2 2 + · · · + x2 n 1}, and Sn−1 = ∂Bn = {x1,x2,...,xn∈ Rn | x2 1 + x2 2 + · · · + x2 n = 1}. We call any space homeomorphic to Bn or Sn−1 an n-cell or an (n − 1)- sphere, respectively. The k-ball Bk is defined as a subset of Rk, but for each k n the inclusion Rk ⊂ Rn determines a standard k-ball Bk and a standard (k − 1)-sphere Sk−1 in Rn as well. All simplicial complexes and CW complexes are assumed to be locally finite. A polyhedron is the underlying space of a simplicial complex. While a simplicial complex K and the underlying polyhedron |K| are two differ- ent things, we will not always maintain this distinction in our terminology. Piecewise linear is abbreviated PL. An n-dimensional (topological) manifold is a separable metric space in which each point has a neighborhood that is homeomorphic to Rn. Such a neighborhood is called a coordinate neighborhood of the point. The Main Problem. The central topic in this text is topological embed- dings. Formally, an embedding of one topological space X in another space Y is nothing more than a homeomorphism of X onto a subspace of Y . The domain X is called the embedded space and the target Y is called the am- bient space. Two embeddings λ,λ : X → Y are equivalent if there exists a (topological) homeomorphism Θ of Y onto itself such that Θ ◦ λ = λ . The main problem in the study of topological embeddings is: Main Problem. Which embeddings of X in Y are equivalent? In extremely rare circumstances all pairs of embeddings are equivalent. For instance, if X is just a point, the equivalence question for an arbitrary pair of embeddings of X in a given space Y amounts to the question of homogeneity of Y , which has an aﬃrmative answer whenever, for example, Y is a connected manifold. Ordinarily, then, our interest will turn to conditions under which embed- dings are equivalent, and we will limit attention to reasonably well-behaved spaces X and Y . Specifically, in this book the embedded space X will ordi- narily be a compact polyhedron1 and the ambient space Y will always be a manifold, usually a piecewise linear (abbreviated PL) manifold. If there are embeddings of the polyhedron X in the PL manifold Y that are homotopic 1 A major exception is the study of embeddings of the Cantor set.

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