1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

182 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS

LEMMA 31.21 (Separating surfaces and incompressibility). Suppose that E^2 is

a closed surface of genus 2: 1 in a closed 3-manifold M^3 separating M into two

compact submanifolds M1 and M2 with 8M1 = 8M2 = M1 n M2 = E. If E

is incompressible in both M 1 and M 2 , then E is incompressible in M. Moreover,

both 7r 1 (M 1 ) and 7r 1 (M2) inject into 7r1(M).

The lemma is an easy consequence of the following:

THEOREM 31.22 (Seifert and Van Kampen). If a topological space X = X1 UX2

is a union of two path-connected open sets X 1 and X2 so that X1 n X2 is path

connected, then

7r1 (X,xo) ~ (7r1(X1, xo)*7r1 (X2, xo)) /H,

where H is the normal subgroup generated by (i1t (a) [(i2t (a)r^1 for all a E

7r1 (X1 nX2,xo) and where

X1 nX2

i1 / \,i2

X1 X2

J I \, /h

X1 UX2

are the inclusion maps. Furthermore, if (i 1 ) and (i2) are injective, then (j 1 )* and

(j2)* are injective.

REMARK 31.23. Many books do not state the last fact in the theorem, which

turns out to be very useful in practice. It is a co nsequence of the normal form

theorem for amalgamated free products of groups; see for instance Theorem 25 in

Chapter 1 of D. Cohen's book [85].

PROOF OF LEMMA 31.21. Let Xa be the union of Ma and an open collar of

E^2 in M for a = 1, 2. Note that E is a deformation retract of X 1 n X 2. Let

xo E E. Since E is incompressible in both M1 and M2, we have that (ia)* :

7r 1 (X 1 n X2, xo) --+ 7f 1 (Xa, xo) is injective for a = 1, 2. So, by Theorem 31.22,

Ua)* : 7r1 (Xa, xo) --+ 7r1 (X1 U X2, xo) = 7f1 (M, xo) is injective for a = 1, 2. The

lemma easily follows. D

REMARK 31.24. For another, "cut and paste", proof of Lemma 31.21, see the

notes and commentary.

2.2. Incompressibility of tori in hyperbolic 3-manifolds.

In the next section (see Theorem 31.44 below) we shall show that each topolog-

ical end of a finite-volume hyperbolic 3-manifold (1i^3 , h) is isometric to [O, oo) x V^2

with the metric

gcusp = d r^2 + e -2r gflat,

where (V, gflat) is a flat torus. Each slice Vr ~ {r} x Vis an embedded flat torus
(with constant second fundamental form) in (1i, h). In this subsection we discuss
topological properties of embedded tori, including Vr, in 1i.

LEMMA 31.25 (Boundary tori of hyperbolic 3-manifolds are incompressible).

Let (1i^3 , h) be a finite-volume hyperbolic 3-manifold. The 2-torus Vr is incompress-

ible in 1i.