2 0. Prequel

Figure 0.1. A bouquet of six circles

Note that

Rk

⊂ R+

n

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

1A

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.