§1. INTRDDUCTIOI

The following theorem is the main result of the paper.

THEOREM 1. Let

N11

be a topological (resp. PL) manifold, A

an ANR and h: Nr + A an embedding. Given an open cover a of A

there exists an open cover g of A such that if wtf + A is a

g-honotopy equivalence from a topological (resp. PL) manifold

onto A, m - n ^ 3, then there exists a locally flat (PL) embed-

ding such that pg is a-close to h. Furthermore,

if m _ 5, given an open cover y of K, vre may choose a fine

enough that any two locally flat (PL) enibeddings 9nr9^' ^ +

M*1/

for which pg. is a-close to h, i = 0,1, are aittoient isotopic

by a (PL) fiber y-push.

The techniques used in the proof of Theorem 1 are radial en-

gulfing arguments together with Miller's approximation techniques and

Bdwards-Kirby handle straightening techniques.

We give three corollaries to Theorem 1.

COROLLARY 1. (Miller, Bryant-Seebeck). Let h: l/1 + #P be a

topological embedding from a topological (resp. PL) manifold

into a topological (resp. PL) manifold m - n 3. Tfoen h can

be arbirarily closely approximated by locally flat (PL) einbeddings.

Received by the editors April 14,1980.

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