Chapter 9. Basic Topology of 3-Manifolds
The purpose of this chapter is to introduce certain well-known facts in
3-manifold topology which are related to the Ricci flow. It is by no means
meant to be a complete survey of the subject, and we have omitted many
important results in the field. Due to space limitations, we will not provide
proofs and instead will refer the reader to the literature.
Unless mentioned otherwise, we shall assume in this chapter that all 3-
manifolds are connected, orientable, and with possibly nonempty boundary.
The n-dimensional sphere, n-dimensional ball and n-dimensional Euclidean
space are denoted by 5n, Bn, and JRn, respectively.
1. Essential 2-spheres and irreducible 3-manifolds
1.1. Topological, PL and smooth categories. The fundamental
work of Moise [268] in 1952 shows that any topological homeomorphism
of an open set in JR^3 into JR^3 can be c^0 approximated by piecewise-linear
(PL) homeomorphisms. As a consequence, he proved that any topological 3-
manifold can be triangulated and that there is a unique PL structure on any
topological 3-manifold. Different proofs of the Moise theorem can be found
in Bing [28] and Shalen [327]. It is shown in Hirsch [203] and Munkres
[279] that the PL and smooth categories in dimension 3 are equivalent. For
the rest of this chapter, we assume that all manifolds and maps between
them are smooth.
1.2. Sphere decompositions and irreducibility. The study of 2-
spheres in 3-manifolds probably began with the Schoenfiies problem. It
asks if any smoothly embedded 2-sphere in Euclidean 3-space must bound a
3-ball. The affirmative solution of the problem in dimension 3 by Alexander
[1] is one of the milestones in the field. We say a 2-sphere in a 3-manifold
is essential if it does not bound a 3-ball. Essential 2-spheres are closely
related to the connected sum decomposition. Indeed, if a 3-manifold Mis a
connected sum M = Mi#M2, where neither Mi nor M2 is 83 or B^3 , then
the decomposing 2-sphere is essential. On the other hand, if 82 is an essential
2-sphere in M which decomposes the manifold into two pieces Ni and N2,
i.e., 82 is separating, then this gives a connected sum decomposition M =
Mi#M 2. Namely, we simply take Mi to be Ni capped off by a 3-ball. If
the essential 2-sphere 82 does not separate M, i.e., M\8^2 is connected, then
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