Luis Mantejano
Corollary 3 follows directly from Theorem 1 when N = A and
h = Id.
In the same way that the Chapman-Ferry Theorem has the Siebermann
Cell-like Approximation Theorem[21] as a corollary, we have as a
consequence of Corollary 3 that codimension three cell-like maps
between manifolds have locally flat approximate cross sections, thus
solving a conjecture posed by Rushing in [20].
In codimension two, we also study the general lifting-problem
dDtaining similar conslusions provided that certain added conditions
are satisfied. In particular we apply Venema's approximation
techniques, [22] and [23] to obtain the following two theorems:
1HEOREM 2. Let A be an ANR , h: D + A an embedding and
U an open subset of D . Given e 0 there exists 6 0 such
that if is a 6-fibration from a PL manifold
onto A such that there exists a locally flat PL embedding
g : U * M with d(pgn,h|u) 6, then there exists a locally flat
PL embedding g: D M with d(pg,h) e.
THEOREM 3. Let w be a coitpact connected 2-manifold with
boundary, A an ANR and h: N -* A an embedding. Given e 0
there exists 6 0 such that if p: M * A is a 6-homotopy equiva-
lence from a PL 4-manifold M onto A, then there exists a local-
ly flat PL embedding g: w * M such that d(pg,h) e.
Similar corollaries to those obtained from Theorem 1 can be
obtained from Theorem 2 and Theorem 3.
Results of Chapman [7] and Edwards [11] allow us to obtain the
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