On embeddings with locally nice cross-sections

Author:
J. L. Bryant

Journal:
Trans. Amer. Math. Soc. **155** (1971), 327-332

MSC:
Primary 57.05

MathSciNet review:
0276983

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Abstract: A -dimensional compactum in euclidean space is said to be locally nice in if is -ULC. In this paper we prove a general theorem which implies, in particular, that is locally nice in if the intersection of with each horizontal hyperplane of is locally nice in the hyperplane. From known results we obtain immediately that a -dimensional polyhedron in ( and ) is tame in if each is -ULC. However, by strengthening our general theorem in the case , we are able to prove this result for as well. For example, an arc in is tame if each horizontal cross-section of is tame in the cross-sectional hyperplane (that is, lies in an arc that is tame in the hyperplane).

**[1]**J. L. Bryant,*On embeddings of compacta in Euclidean space*, Proc. Amer. Math. Soc.**23**(1969), 46–51. MR**0244973**, 10.1090/S0002-9939-1969-0244973-1**[2]**J. L. Bryant and C. L. Seebeck III,*Locally nice embeddings of polyhedra*, Quart. J. Math. Oxford Ser. (2)**19**(1968), 257–274. MR**0234434****[3]**J. L. Bryant and C. L. Seebeck III,*Locally nice embeddings in codimension three*, Quart. J. Math. Oxford Ser. (2)**21**(1970), 265–272. MR**0290376****[4]**Herman Gluck,*Embeddings in the trivial range*, Ann. of Math. (2)**81**(1965), 195–210. MR**0173243****[5]**C. D. Papakyriakopoulos,*On Dehn’s lemma and the asphericity of knots*, Ann. of Math. (2)**66**(1957), 1–26. MR**0090053**

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DOI:
https://doi.org/10.1090/S0002-9947-1971-0276983-6

Keywords:
Locally nice embeddings,
-ULC subsets of ,
tame embeddings,
embeddings with tame cross-sections,
embeddings with locally nice cross-sections,
topological embeddings of compacta,
topological embeddings of polyhedra

Article copyright:
© Copyright 1971
American Mathematical Society