2 0. Prequel
Figure 0.1. A bouquet of six circles
if k n.
An n-dimensional ∂-manifold (read “boundary manifold”) is a separable
metric space in which each point has a
that is homeomorphic
We will use superscripts to denote the dimension of a manifold or a
∂-manifold. Thus the statement “M
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
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.
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
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.