1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

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  1. NOTES AND COMMENTARY 195


For a proof of the Jordan- Brouwer separation theorem, seep. 89 of Guillemin
and Pollack [131].
Using a standard low-dimensional topological "cut and paste" argument, we
give a self-contained proof of Lemma 31.21.
We successively replace a map J of a disk into a 3-manifold by homotopic maps
with better properties (see for example Scott and Wall [360]). Suppose that

Then, by Corollary 31. 15 , there exists a E ker{i*: 7r 1 (E)--+ 7r 1 (M)} - {l} such
that there exists a smooth embedding of a closed disk J : D^2 --+ M such that^10

J(8D) CE and [JlaDl =a.


By deforming J in a thin collar of 8D in D , we may assume that J is "nor-
mal" (in particular, transversal) to E in this collar of 8D. Then, by applying the
Transversality Homotopy Theorem (see p. 70 of [131]) to J restricted to D minus
a thinner collar of 8D, we may assume that the map Jlint(D) is transversal to E
without changing Jin this thinner coll ar. This implies that J-^1 (E) CD is a codi-
mension 1 submanifold.11 Since J is transversal to E in a collar of 8D, we have
J-^1 (E) is a finite disjoint union of loops, where each loop is contained in int (D)
unless the loop is equal to 8D itself.
Now consider an "innermost" loop C C J-^1 (E), i.e., a lo op C whose interior
does not contain any loops of J-^1 (E). Then J(C) c E and C bounds a disk
B^2 c D whose image under J is contained in either M 1 or M 2. In either case,
by our assumption that E is incompressible in both M 1 and M 2 , we can replace
JIB by a homotopic map sending B entirely into E. Keeping J the same outside of

B , this yields a new map from D homotopic to the original, which we still call f.


We may then slightly push Jin a neighborhood of B to the other side (of JIB) to
remove C from J-^1 (E). Repeating this process at most a finite number of times
eventually yields no loops in J-^1 (E) except 8D. That is, we may assume that

either J (D) C M 1 or J (D) C Mz. Again using that E is incompressible in both


M 1 and M 2 , we conclude that JlaD must be null-homotopic in E. This yields a
contradiction and hence shows that 7r 1 (E) injects into 7r 1 (M).
A similar argument shows that both 7r 1 (M 1 ) and 7r 1 (M 2 ) inject into 7r1(M).
We leave it as an exercise for the reader to verify this.


§3. For Theorem 31.36, see Theorem 5.10.l in [ 401 ] or Theorem D .1.1 in [24].
For a proof of Theorem 31.38, see Proposition D.2.1 and Theorem D.2.2 of [24].
For a proof of Lemma 31.42, see Proposition D.3.2 of [24].
For a proof of Theorem 31.43, see Theorem D.3.3 and Proposition D.3.12 of
[24].
For a proof of Theorem 31.44, see Proposition D.3.12 on pp. 151 - 1 52 of [24].
For Example 31.45, seep. 151 of [24].

(^10) Note that Corollary 31.15 holds in the smooth category.
(^11) In general, if F: X -7 Y is transversal to Z C Y , then p-l (Z) is a submanifold of X ,
where the codimension of p-l (Z) in X is equal to the codimension of Z in Y (see the theorem
on p. 28 of [131]).

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