2 0. Prequel Figure 0.1. A bouquet of six circles Note that Rk ⊂ Rn + if k n. An n-dimensional ∂-manifold (read “boundary manifold”) is a separable metric space in which each point has a neighborhood1 that is homeomorphic to R+. n We will use superscripts to denote the dimension of a manifold or a ∂-manifold. Thus the statement “M n is a manifold” is to be interpreted to mean that M is an n-dimensional manifold. Let M be an n-dimensional ∂-manifold. The interior of M (denoted Int M) consists of all points x ∈ M such that x has a neighborhood that is homeomorphic to Rn. The boundary of M (denoted ∂M) is defined by ∂M = M Int M. Remark. Our use of the term ∂-manifold is somewhat nonstandard, but we prefer it to the more awkward manifold-with-boundary. The use of the term ∂-manifold allows us to be consistent in our use of the word manifold: in this book a manifold always has empty boundary. A closed manifold is a manifold that is compact and has empty boundary. Since all our manifolds have empty boundary (by definition), there is no difference between a closed manifold and a compact manifold. The Invariance of Domain Theorem (Munkres, 1984, Theorem 4-36.5) should be used to work several of the following exercises. Exercises 0.1.1. The dimension of a manifold is well defined: two manifolds of dif- ferent dimensions cannot be homeomorphic. 0.1.2. The dimension of a ∂-manifold is well defined. 0.1.3. Let M be an n-dimensional ∂-manifold and let y be a point in M. If the last coordinate of h(y) is zero for one pair (U, h) in which U 1 A neighborhood is not necessarily an open set. A neighborhood of the point x in the space X is any subset U of X such that x is contained in the topological interior of U.

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