B-homotopy equivalences 5
following codiniension two stable version of Corollary 3.
THEOREM 4. Let be a topological manifold, n ^ 2. Given an
open cover a of N there exists an open cover 3 of N such that
is a topological manifold and is a proper
g-hcmotopy equivalence with p|8M: 3M - N a locally trivial bundle,
then for every open cover y of E
pxl:Mxl-Nx]R has a
(a x y)-cross section g: Nxi-MxE.
Finally we obtain the following codimension one cross section
THEOREM 5. Let N11 be a topological manifold, n 4. Given an
open cover a of N there exists an open cover g of N such that
if lyP"1 is a topological manifold and + IN is a proper (Miono-
topy equivalence with p|9M: 8M - N a locally trivial bundle, then
there exists a locally flat a-cross section g: N - * M.
The proof of Theorem 5 uses results of Chapman in  and of
I wish to express ny gratitude and appreciation to T.B. Rushing
for his valuable help and advice with this research. I also want to
thank Leticia for her constant encouragement.