Somewhere locally flat codimension one manifolds with complements are locally flat

Authors:
T. M. Price and C. L. Seebeck

Journal:
Trans. Amer. Math. Soc. **193** (1974), 111-122

MSC:
Primary 57A45

DOI:
https://doi.org/10.1090/S0002-9947-1974-0346796-8

MathSciNet review:
0346796

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to prove a taming theorem for a codimension one manifold that is locally flat at some point and has 1-ULC complement. We also prove that any two sufficiently close locally flat embeddings of a codimension one manifold are ambient isotopic. Since this paper was first submitted, R. Daverman has shown that, given any point on a codimension one manifold with 1-ULC complement, some neighborhood of that point lies on a codimension one sphere that is locally flat at some points and has 1-ULC complement. Hence the two papers combined prove that a codimension one manifold is locally flat if and only if its complement is 1-ULC.

**[1]**J. L. Bryant,*Euclidean n-space modulo an*-*cell*, Trans. Amer. Math. Soc. 179 (1973), 181-192. MR**0324703 (48:3053)****[2]**A. V. Černavskiĭ,*The k-stability of homeomorphisms and the union of cells*, Dokl. Akad. Nauk SSSR 180 (1968), 1045-1047 = Soviet Math. Dokl. 9 (1968), 729-732. MR**37**#6919. MR**0231364 (37:6919)****[3]**-,*Locally homotopic unknotted imbeddings of manifolds*, Dokl. Akad. Nauk SSSR 181 (1968), 290-293 = Soviet Math. Dokl. 9 (1968), 835-839. MR**38**#720. MR**0232395 (38:720)****[4]**R. D. Edwards and R. C. Kirby,*Deformations of spaces of imbeddings*, Ann. of Math. (2) 93 (1971), 63-88. MR**44**#1032. MR**0283802 (44:1032)****[5]**R. C. Kirby and L. C. Siebenmann,*On the triangulation of manifolds and the hauptvermutung*, Bull. Amer. Math. Soc. 75 (1969), 742-749. MR**39**#3500. MR**0242166 (39:3500)****[6]**C. L. Seebeck III,*Collaring an*-*manifold in an n-manifold*, Trans. Amer. Math. Soc. 148 (1970), 63-68. MR**41**#2692. MR**0258045 (41:2692)****[7]**P. Wright,*A uniform generalized Schoenflies theorem*, Ann. of Math. (2) 89 (1969), 292-304. MR**40**#3556. MR**0250317 (40:3556)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57A45

Retrieve articles in all journals with MSC: 57A45

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0346796-8

Keywords:
Codimension one,
homeomorphic approximation,
1-ULC complement,
radial engulfing

Article copyright:
© Copyright 1974
American Mathematical Society