1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. INCOMPRESSIBLE SURFACES AND GEOMETRIZATION CONJECTURE 435

was generalized by W. Haken to normal surface theory [1 76].) Let us fix

a triangulation T of a compact 3-manifold M. Let t be the number of

tetrahedra in the triangulation. Kneser proved that if n > 6t + I H 1 ( M; Z2) [,

where H1 (M; Z2) is the first homology group of M with Z 2 coefficients,

then any n disjoint essential 2-spheres contain a parallel pair. Here is the

basic argument. Suppose { S1, ... ,Sn} is a collection of n essential disjoint

2-spheres in M. By isotopy and topological surgeries, one may find a new

collection of n essential disjoint 2-spheres, still denoted by { S 1 , ... ,Sn}, that

are in 'nice position' with respect to the triangulation. Here, 'nice position'

means that the intersection of each 2-sphere with each tetrahedron consists

of a disjoint union of geometric triangles and quadrilaterals. These are

called normal surfaces. There are only seven normally isotopic triangles

and quadrilaterals in a tetrahedron. This shows that inside a tetrahedron

0'^3 , all but at most six components of 0"^3 \(S1 U · · · USn) are parallel regions.

It follows from this fact by a. simple computation that there are two parallel

2-spheres if n > 6t+ [H1(M;Z2)[.

1.3. Irreducible 3-manifolds. Irreducible 3-manifolds joined in con-

nected sums may be regarded as building blocks for 3-manifolds. Most

familiar 3-manifolds are irreducible. For instance, the complement of a knot

in S^3 is irreducible by Alexander's theorem. Also, if a covering space N of

a 3-manifold M is irreducible, then M is irreducible. This is due to the

fact that 2-spheres are simply connected. Thus any 2-sphere in M can be

lifted to a 2-sphere in N. Now using the irreducibility of N, one produces

a 3-ball in N bounding the lifted 2-sphere. Using the Brouwer fixed point

theorem, one then shows that the 3-ball is mapped injectively to M by the
covering map. This proves M is irreducible. In particular, if the universal
cover of a 3-manifold is ffi.^3 , then the manifold is irreducible. This shows, for
instance, that all fiat (e.g., S^1 x S^1 x S^1 ) and all hyperbolic 3-manifolds are
irreducible. A deep result of Meeks and Yau [263] shows that the converse
is also true. Namely, if a manifold is irreducible, then all covering spaces of
it are irreducible.
In [264] Milnor proved that the connected sum decomposition of a com-
pact 3-manifold is unique up to self-homeomorphism of the 3-manifold. But
the decomposition in Kneser's theorem is in general not unique up to iso-
topy of the 3-manifold. This is due to the action of the diffeomorphism
group of the 3-manifold on the decomposition. For instance, the manifold

(S^2 x S^1 )#(S^2 x S^1 ) has many nonisotopic essential separating 2-spheres,

due to the large diffeoinorphism group of the manifold.

2. Incompressible surfaces and the geometrization conjecture

2.1. Incompressible surfaces and Haken manifolds. The success

of the study of 2-spheres in 3-manifolds prompted people to look for more
general surfaces. Evidently, surfaces that can be contained inside a coordi-
nate chart are not going to be interesting. Haken introduced the following