Let M"1, N11 be manifolds, A an ANR, a surgective
map, h: N - A an embedding and a an open cover of A. The
purpose of this paper is to study the following problem: Vlhen is
there a locally flat embedding g: such that pg is
a-close to h? Our answers are phrased in terms of homotopy
restrictions on p. This problem is intimately related to the
problem of approximating topological errbeddings with locally flat
Library of Congress Cataloging in Publication Data
Montejano, Luis, 1951-
fi -homotopy equivalences have e-cross sections.
(Memoirs of the American Mathematical Society,
ISSN 0065-9266 ; no. 27*0
1. Homotopy equivalences. 2, Embeddings (Mathematics)
I. Title. II. Title: Beta-homctopy equivalences have
alpha-cross sections. III.Series.
QA3.A5T no. 27^ CQA.6l2.723 510s C5lUf.2l+] 82-206l6
1980 Mathematics Subject Classification. Primary 57C55, 57A60;
Secondary 57A30, 57A35.
Key words and phrases: cell-like maps, cell-like enbedding relations,
3-hcmotoy equivalences, engulfing, approximations.
1"The material in this paper will beincluded in the author's Ph.D.
dissertation written under the direction of Professor T.B. Rushing at
the University of Utah. Research partially supported by a University
of Utah Graduate Research Fellowship and the National University of