1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. TOPOLOGY AND GEOMETRY OF HYPERBOLIC 3-MANIFOLDS isi

7ri (E^2 ) Iker (i*) must be isomorphic to either 0, z, or z x Z.^2 By the classification

of abelian groups, ker ( i*) is isomorphic to Z provided ker ( i*) =f. ni (E).

On the other hand, by Corollary 31.15, a nontrivial element of ker (i*) is rep-

resented by a simple closed curve in E. Since this element must be primitive, it

generates ker (i*) ~ Z. 0

For the statement and proof of the next result, all homology and cohomology

coefficients are JR.

LEMMA 31.20 (Induced map on Hi for the boundary inclusion). Let M^3 be a

compact orientable 3 -manifold with boundary E^2 = 8M, which may be disconnected.

Let i : E Y M be the inclusion map. The rank of

i* : Hi (E) ---+ Hi (M)

is equal to half the dimension of Hi (E).

PROOF. Recall the following two basic facts from linear algebra. If A : !Rn ---+

!Rm is a linear map and if A* : !Rm ---+ !Rn denotes its dual, then

(31.7) dim(image(A)) = dim(image(A*)) = rank(A)

and

(31.8) dim (ker (A))+ dim (image (A)) = n.

We have the short exact (co )homology sequence

(31.9)

Hi (M) ~ Hi (E),

where D is the isomorphism from Lefschetz duality (this requires E = 8M) and

where i is the dual of i. Thus, by (31.7), we have

dim (image (j)) =dim (image (i)) =dim (image (i)).

Now the exactness of the sequence in (31.9) says

dim (image (j)) =dim (ker (i*)),

so that

dim (image (i)) =dim (ker (i)).

Finally, by (31.8), we have

dim (ker (i)) +dim (image (i)) =dim (Hi (E)),

so that dim (ker (i*)) =~dim (Hi (E)).^3 0

We have the following standard results from geometric topology, which we shall

use in the next subsection.

(^2) That is , we have ruled out the possibilities llm, Zn x Zm, and Z x Zn.

3Note that the above information on the kernel of i. : H1 (~) -t H1 (M), on the level of

homology, says nothing about the kernel of i. : rr1 (~) -t rr1 (M).