10 Luis Mbntejano
THEOREM 3.5. Given e 0 there exists 0 6 e such that
if:
a) p: M +3R is a surjective
map,
b) S is a combinatorial suhmanifold of M with
mesh(S)
p"1^)
,
c) C c
p"1(w (Dk))
n s is a subocmplex of S ,
d) C is a (P/j/6)-collapse, and
e)
G3"1: D^"1
- 9N(C,S") is a PL embedding with
d(pGD" (x),x) 6 for every D11"1 ,
thenthere exists a PL embedding
G-5: D-1
- N(C,S") with
d(pG^(x),x) e for every x e D- * .
PROOF. The proof is virtually identical to the proof of
Theorem 8 in [17].
DEFINITION 3.6. A sequence of subccmplexes of M
{c ,..., CP}{B ,..., B^} is called a (p,e)-sequence if:
-1 k
1) There exists a triangulation T, of p (N (D )) with
mesh(Tk) p" (e) ,
2) Ck c p-1(N (Dk)) is a subccmplex of T, ,
3) c is a (p,kr£)-collapse and dim CT k
r
4)
Bk
= N(C\T£) ,
5) There exists a triangulation T of 9B n p" (N
(Dr))
with mesh(Tr) p^Ce) , j r k - 1
f
6) cf c 3Br+1 n p""1(N (Dr)) is a subocmplex of T ,
j r k - 1 ,
7) CT is a (p,r,e)-collapse and dim CT r, j r k ,
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