g-homotopy equivalences
By section 6 of [3] , we may assume,without loss of general-
ity, that R is a 1-LCC cell-like embedding relation, consequently,
A has the disjoint disk property. Then, by Edwards's Cell-like
Approximation Theorem [11], the map p: M - A is a near hcmecmor-
phism. The inverses of the homeomorphisms that approximate p ,
restricted to pR(N) , give rise to the desired embeddings.
Originally, we were interested in the special case of Theorem 1
where A = N and h = Id . Explicitly, if p: ^ N
1 1
is a
surjective map, when does p have an a-cross section? (i.e. when
is there an anbedding such that pg is a-close to
IcL ?) . The first result in this direction was established by
Chapnan and Ferry in [8] and corresponds to the codimension zero case
of this problem. They proved that proper g-hcmotopy equivalences
between manifolds of the same dimension can be a-approximated by
hamecmorphisms and consequently have a-cross sections.
We obtain the following codimension three analogue of that
COROLLARY 3. Let / be a topological (resp. PL) manifold.
Given an open cover a of N there exists an open cover g of N
such that if is a g-homotopy equivalence f rem a topolo-
gical (resp. PL) nt-manifold onto m - n 3, then there
exists a locally flat (PL) a-cross section g: N - * M. Furthermore,
if m 5, given an open cover y °f
Y choose a fine
enough that any two locally flat (PL) a-cross sections in M are
(PL) ambient isotopic by a fiber Y-push.
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