§3. CODIMENSION THREE LOCAL CROSS SECTIONS
The prupose of the next two sections is to prove, using the
techniques of R.T. Miller in , 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
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:
a surjective map, m - k 3.
Let O^i 3K +1^ be the following hatotopy: 1 j k.