2. TOPOLOGY AND GEOMETRY OF HYPERBOLIC 3 -MANIFOLDS 179
PROOF. Let 52 be an embedded 2-sphere in a complete hyperbolic manifold
( 1-l^3 , h) and let p : JHI^3 --+ 1-l be the universal covering. Since 52 is simply connected,given x E 52 and a choice of x E p-^1 (x ), there exists a unique lift 5
2c JHI^3 with
-2 -2 -2x E 5 , where 5 is an embedded 2-sphere. By the Schonflies theorem, 5 bounds
an embedded 3-ball B^3 c JHI^3. We claim that B^3 ~ p(B) is an embedded 3-ball in 1-l
bounding 52. Indee d , Pis : B --+ Bis a local diffeomorphism with Pl.s2 : 5
2
--+ 52 a
homeomorphism. Hence, by degree theory, Pis is 1-1 and hence a diffeomorphism;
the claim follows. DRecall that a surface of genus g ;::: 1 embedded in a 3-manifold by l : E^2 '---+ M^3
is incompressible if the map induced on fundamental groups L. : 7T 1 (E) --+ 7T 1 (M)
is injective. If E is not incompressible in M, then we say that E is compressible inM. In the case where E ~ 52 , one says that an embedded 52 c M is compressible
if it bounds a 3-ball in M. (We have actually used an equivalent definition of
incompressibility; see pp. 31-34 of Jaco [158].)
Let D^2 denote the closed unit ball in JR^2.THEOREM 31.14 (Loop Theorem). LetM^3 be a compact3-manifold, letE^2 be a
connected surface in 8M, and let N c 7T 1 (E) be a normal subgroup. If ker (i.)-N
is nonempty, where i : E '---+ M is the inclusion map, then there exists an embedding
j : D^2 --+ M such that
j (8D) CE and [Jl 8 DJ tf_ N ,where [jl 8 D] E 7T1 (E) denotes the homotopy class of JlaD·
This result, due to Papakyriakopoulos, is useful for studying the compressibility
and incompressibility of embedded surfaces in 3-manifolds. Taking N = {1}, we
have the following.
COROLLARY 31.15 (Compressible boundary components). If M^3 is a compact
3-manifold and E^2 is a closed connected two-sided compressible surface of genus;::: 1 in 8M, then there exists an element 1 #-a E ker (i.), where i : E^2 '---+ M is
the inclusion map, and an embedding j : D^2 --+ M such thatj (8D) c E and [Jl 8 D] =a.
In particular, a may be represented by a simple closed curve in E , which implies a
is a primitive element (i.e., it is not a nontrivial power of another element).
In the last sentence we used the following.
LEMMA 31.16. Any element of the fundamental group of an oriented closed
surface which is represented by a simple closed curve is a primitive element.
PROOF. Suppose that (}, with [(}] E 7r 1 (E,p) - {id}, is a simple loop in an
oriented closed surface E^2. By the classification of surfaces, E is homeomorphic
to the compact plane region bounded by a 4g-sided polygon with sides labeled
a 1 ,b 1 ,a!\b1^1 ,.. .,a 9 ,b 9 ,a;^1 ,b;^1 and identified accordingly, where g 2 1. In E
each of these sides, ai or bj, identifies to a nonseparating loop; i.e., E - ai and
E - bj are connected for all i,j = 1, ... , g.
(1) Suppose that (} is not separating. We claim that there exists a self-
homeomorphism of E mapping (} to a 1. To see this, let Ea and Ea 1 be the connected
compact surfaces with two boundary circles obtained by cutting E along (} and al,