436 9. BASIC TOPOLOGY OF 3-MANIFOLDS
important concept. A compact, connected, properly embedded surface F
in a 3-manifold M is said to be incompressible if one of the following
conditions holds: ·
(1) F i= 52 or B^2 and the inclusion map induces an injective homo-
morphism in the fundamental group; or
(2) F = 52 is an essential 2-sphere; or
(3) F = B^2 and 8F is not null homotopic in 8M.
An orientable Haken 3-manifold is a compact and irreducible 3-mani-
fold M that admits a two-sided incompressible surface. This definition is
the same as saying that M is a compact, orientable, and irreducible mani-
fold which contains an incompressible surface other than JR.JED^2. The reason
is that if a compact, irreducible, orientable 3-manifold M contains U^2
as an incompressible surface, then M = ffi.JED^3. Also, if a compact surface
is incompressible in M and is one-sided, then the boundary of a regular
neighborhood of the surface is a two-sided incompressible surface. Further-
more, since we assume the 3-manifold is orientable, two-sided surfaces are
the same as orientable surfaces.
Haken manifolds constitute a huge portion of all of 3-manifolds. For in-
stance, if a compact orientable 3-manifold Mis irreducible and has nonempty
boundary, then M is Haken. Also, if a closed, irreducible 3-manifold has
positive first Betti number, it is Haken. The homeomorphism classification
of Haken manifolds is considered to be solved. It is due to the deep work
of F. Waldhausen [363] in 1968. Among the many results he proved, the
following stands out as one of the most striking.
THEOREM 9.2 (Waldhausen). Two homotopically equivalent closed Haken
manifolds are homeomorphic.2.2. Torus decompositions and the geometrization conjecture.
A Seifert 3-manifold (also called a Seifert space) is a compact 3-manifold
admitting a foliation whose leaves are 51. Although this was not the original
definition by Seifert in 1931, subsequent work of Epstein [136] shows that
this simpler definition is equivalent to Seifert's original formulation. If a
compact 3-manifold admits an 51 action without global fixed points (i.e.,
no point is fixed by all elements in 51 ), then the manifold is a Seifert space.
Seifert 3-manifolds have been classified. In particular, there exist Seifert
manifolds which are irreducible, non-Haken 3-manifolds and have infinite
fundamental group. In 1976, Thurston constructed closed hyperbolic 3-
manifolds that are not Haken.
Suppose Mis a closed, irreducible, and orientable 3-manifold. A natural
step after the connected sum decomposition is to look for incompressible tori.
This is called the torus decomposition. A compact, irreducible 3-manifold
is called geometrically atoroidal if every incompressible torus is isotopic
to a boundary component. (If the manifold is closed, this simply means that
there are no incompressible tori.) The torus decomposition theorem of Jaco