Singular regular neighborhoods and local flatness in codimension one

Author:
Robert J. Daverman

Journal:
Proc. Amer. Math. Soc. **57** (1976), 357-362

MSC:
Primary 57A45

MathSciNet review:
0420630

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For an -manifold topologically embedded as a closed subset of an -manifold , we define what it means for to have a singular regular neighborhood in . The principal result demonstrates that has a singular regular neighborhood in if and only if the homotopy theoretic condition holds that is locally simply connected (-LC) at each point of . Consequently, has a singular regular neighborhood in if and only if is locally flatly embedded .

**[1]**R. H. Bing,*A surface is tame if its complement is 1-ULC*, Trans. Amer. Math. Soc.**101**(1961), 294–305. MR**0131265**, 10.1090/S0002-9947-1961-0131265-1**[2]**J. W. Cannon,*𝑈𝐿𝐶 properties in neighbourhoods of embedded surfaces and curves in 𝐸³*, Canad. J. Math.**25**(1973), 31–73. MR**0314037****[3]**A. V. Černavskiĭ,*The equivalence of local flatness and local -connectedness for imbeddings of -dimensional manifolds in -dimensional manifolds*, Mat. Sb.**91 (133)**(1973), 276-286 = Math. USSR Sbornik**20**(1973), 297-304.**[4]**Robert J. Daverman,*Locally nice codimension one manifolds are locally flat*, Bull. Amer. Math. Soc.**79**(1973), 410–413. MR**0321095**, 10.1090/S0002-9904-1973-13190-8**[5]**Samuel Eilenberg and Norman Steenrod,*Foundations of algebraic topology*, Princeton University Press, Princeton, New Jersey, 1952. MR**0050886****[6]**D. B. A. Epstein,*The degree of a map*, Proc. London Math. Soc. (3)**16**(1966), 369–383. MR**0192475****[7]**John Hempel,*A surface in 𝑆³ is tame if it can be deformed into each complementary domain*, Trans. Amer. Math. Soc.**111**(1964), 273–287. MR**0160195**, 10.1090/S0002-9947-1964-0160195-7**[8]**N. Hosay,*The sum of a real cube and a crumpled cube is*, Notices Amer. Math. Soc.**10**(1963), 666; Errata, ibid.**11**(1964), 152. Abstract #607-17.**[9]**Witold Hurewicz and Henry Wallman,*Dimension Theory*, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR**0006493****[10]**Lloyd L. Lininger,*Some results on crumpled cubes*, Trans. Amer. Math. Soc.**118**(1965), 534–549. MR**0178460**, 10.1090/S0002-9947-1965-0178460-7**[11]**M. H. A. Newman,*Local connection in locally compact spaces*, Proc. Amer. Math. Soc.**1**(1950), 44–53. MR**0033530**, 10.1090/S0002-9939-1950-0033530-3**[12]**Paul Olum,*Mappings of manifolds and the notion of degree*, Ann. of Math. (2)**58**(1953), 458–480. MR**0058212****[13]**T. M. Price and C. L. Seebeck III,*Somewhere locally flat codimension one manifolds with 1-𝑈𝐿𝐶 complements are locally flat*, Trans. Amer. Math. Soc.**193**(1974), 111–122. MR**0346796**, 10.1090/S0002-9947-1974-0346796-8**[14]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****[15]**Warren White,*Some tameness conditions involving singular disks*, Trans. Amer. Math. Soc.**143**(1969), 223–234. MR**0248790**, 10.1090/S0002-9947-1969-0248790-2**[16]**J. L. Bryant and R. C. Lacher,*Embeddings with mapping cylinder neighborhoods*, Topology**14**(1975), 191–201. MR**0394680**

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

Retrieve articles in all journals with MSC: 57A45

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0420630-7

Keywords:
Singular regular neighborhood,
locally singularly collared,
-ULC embedding,
locally flat embedding,
degree one mapping of manifolds

Article copyright:
© Copyright 1976
American Mathematical Society