1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

434 9. BASIC TOPOLOGY OF 3-MANIFOLDS

one can again obtain a connected sum decomposition M = (5^2 x 5^1 )#M'.

This gives more information than a separating 2-sphere.

Here is a way to see the 52 x 51 factor in M. Take an embedded arc

A whose endpoints lie in 52 in such a way that A starts from one side

of 52 , ends at the other side, and has no other intersection with 52. We

may assume that, after an isotopy, the endpoints of A are the same. Thus

there is an embedded 51 in M which intersects the 2-sphere transversely

in one point. Let N be a small regular neighborhood of 52 U 51. Then it

is easy to see that N is homeomorphic to ( 52 x 51 ) \ { open 3-ball }. In

particular, this shows that the boundary of N is a 2-sphere. This gives the

connected sum decomposition. (To see the topology of N, the reader may

try to consider the corresponding problem in dimension two: replace 52 by
51 inside a surface. In this case, we have two simple loops a and b inside an
oriented surface so that a intersects b in one point transversely. Then it is
an elementary exercise in topology to show that the regular neighborhood
N (a U b) is ( 51 x 51 ) \ { open 2-disk }. It turns out that this fact holds in all
dimensions.)
Thus essential 2-spheres correspond to connected sum decompositions.
A 3-manifold is called irreducible if each embedded 2-sphere bounds a 3-
ball. If a 3-manifold M is not irreducible, it contains an essential 2-sphere.
Using the operations above, one concludes that either M = 52 x 51 or

M = M1#M2, where neither M1 nor M2 is 53 or B^3. Now one asks

if each of the factor 3-manifolds is irreducible or not. Continuing in this

way, one bumps into the question of whether this decomposition process

for a compact 3-manifold stops after finitely many steps. This was resolved

affirmatively by Kneser in 1929 [233] for compact triangulated 3-manifolds.

THEOREM 9.1 (Kneser). Let M^3 be a compact triangulated 3-manifold.

Then M can be decomposed into a connected sum

Mf#M~# · · · #M~#(5^2 x 51 )# · · · #(5^2 x 51 ),

where each Mi is irreducible.

A counterexample to the Poincare conjecture is usually called a 'fake'
3-sphere. An interesting consequence of Kneser's Theorem is the following
statement: if there exists a counterexample to the Poincare conjecture in
dimension 3, then there exists an irreducible fake 3-sphere. This holds be-
cause the fundamental group of a connected sum is the free product of the
fundamental groups of the factors.
One calls two 2-spheres in a 3-manifold parallel if they are disjoint and
bound a region homeomorphic to 52 x [O, l]. Kneser's theorem states that
for any compact triangulated 3-manifold, there is an integer k such that any
collection of more than k disjoint essential 2-spheres in the manifold must
contain a pair of parallel 2-spheres. Since Kneser's finiteness theorem is so
deeply related to Hamilton's program, we will indicate the basic ideas of
its proof here. (See [201] for a complete proof. Note too that the proof