g-homotopy equivalences

3

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

result.

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

N

we

m

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.