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.