442 9. BASIC TOPOLOGY OF 3-MANIFOLDS
After the surgery, one continues the Ricci fl.ow on the resulting (possi-
bly disconnected) manifold, taking the glued (smoothed) metric as initial
data. Heuristically, this surgery procedure corresponds to Kneser's sphere
decomposition theorem. It is known that a 2-sphere removed from the neck
may bound a 3-ball. Thus for an arbitrary initial 3-manifold M, there is no
guarantee that this surgery process will stop in finitely many steps.
The finiteness theorem of Kneser is in some sense related to a finiteness
conjecture of Hamilton for Ricci fl.ow. The latter states that if one runs
the unnormalized Ricci fl.ow on a closed 3-manifold and performs geometric-
topological surgeries whenever the Ricci fl.ow develops a finite time singu-
larity, then after finitely many such surgeries, the Ricci fl.ow will have a
nonsingular solution for all time. (Since one discards 53 and 52 x 51 fac-
tors, this solution may well be empty.) Using Kneser's finiteness theorem,
one sees that if Hamilton's conjecture does not hold, then all but finitely
many geometric-topological surgeries at finite time singularities of the Ricci
fl.ow must split off 3-spheres. (In this regard, see the following recent pa-
pers: Perelman [299] and Colding and Minicozzi [116].) In this context,
Hamilton's finiteness conjecture may be regarded as a geometric refinement
of Kneser's theorem.
Another type of Ricci fl.ow behavior on 3-manifolds reflects the torus
decomposition. This was first noticed in the work of Hamilton [190]. For
simplicity, we recall Hamilton's formulation. He assumes that a solution
to the normalized Ricci fl.ow on a closed 3-manifold M exists for all time
t E [O, oo) with uniformly bounded curvature. In this scenario, as time ap-
proaches infinity, it may happen that a manifold M can be decomposed into
two parts M = Mthin U Mthick· In the components of Mthin, the metrics
are collapsing with bounded curvature. (Recall that a manifold is said to
be collapsible if it admits a sequence of Riemannian metrics of uniformly
bounded curvature and volumes tending to zero.) In the components of
Mthick, the metrics converge to complete hyperbolic metrics of finite vol-
ume. Furthermore, using minimal surface techniques, Hamilton proves that
the fundamental group of Mthin n Mthick injects into the fundamental group
of M. By the work of Cheeger and Gromov on collapsing manifolds, one
concludes that Mthin is a Cheeger-Gromov graph manifold and hence a
connected sum of Seifert spaces and sol-geometry manifolds. As a conse-
quence, Hamilton was able to establish the geometrization conjecture under
the (restrictive) hypotheses of long-time existence and uniformly bounded
curvature. Perelman's recent work [298], in conjunction with Shioya and
Yamaguchi [332], claims to establish a similar picture without the assump-
tion of bounded curvature.
4. Notes and commentary
We would like to thank Ian Agol for carefully reading this chapter and
for pointing out mistakes in a draft version. We also note that Milnor's