§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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