§3. CODIMENSION THREE LOCAL CROSS SECTIONS

The prupose of the next two sections is to prove, using the

techniques of R.T. Miller in [17], the following two theorems:

THEOREM 3.1. Let A be an ANR and h: D - * A an embedding.

Given e 0 there exists 6 0 such that if is a

6-horrDtopy equivalence (resp. 6-fibration) from a topological

(resp. PL) manifold jyP onto A, m - k 3, then there exists a

locally flat (PL) embedding g: D - jyT with the property that

d(pg,h) e.

THEOREM 3.2. Let k 0 be an integer. Given e 0 there

exists 6 0 such that if -]R is a 6-fibration from a

PL manifold onto Euclidean k-space, m - k 3, then there

exists a PL embedding g: D • M with the property that

d(pg(x),x) e for every xe D .

Theorem 3.2 is a special case of Theorem 3.1. It will be easy

to see that the techniques used in the proof of Theorem 3.2 can be

easily adapted to prove Theorem 3.1 when M is a PL manifold. For

simplicity, we will prove in detail only Theorem 3.2 and at the end

of section 4, we will indicate how to prove Theorem 3.1 when M is

a topological manifold.

In sections 3 and 4, M™ is a PL-manifold and p:

if0,

-]R is

a surjective map, m - k 3.

Let O^i 3K +1^ be the following hatotopy: 1 j k.

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