2 Luis Montejano
Furthermore, if m 5, given an open cover y of M there exists
an open cover a of M such that any two locally flat (PL) embed-
o
dings 9n'9i: N "* " M' which are a-close to h, are (PL) ambient
isotopic by a y-push.
COROLLARY 2. Let R: N31 -+ tfP be a proper cell-like embedding
relation from a topological (resp. PL) manifold w into a topolo-
gical (resp. PL) manifold M , m - n 3. Then R can be arbitrarily
closely approxiraated by locally flat (PL) ertibeddings. Furthermore,
if m 5, given a neighborhood V of R in N x M there exists a
neighborhood W of R in V such that if G^G^: N - * fi are locally
flat (PL) embeddings contained in W, then there exists a (PL)
ambient isotopy H : M * M with the property that H.GQ e V for
every t e I and H..G0 = G,.
Our proof of Corollary 2 goes as follows: Let A be the
decomposition space arising from the upper-seraicontinuous decomposi-
tion of M into point images of R and singletons of M - R(N) .
By [15] A is a finite dimensional ANR . Let p: M + A be the
associated cell-like projection map. By [13], p is an
a-homotopy equivalence for every open cover a of A . Since
pR: N - * A is an embedding, Corollary 2 follows directly from
Theorem 1.
A completely different proof of Corollary 2 can be derived from
work of Ancel-Cannon and Edwards. This results follows from the
following trick which was independently noticed by Jim Cannon.
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