Approximating embeddings of polyhedra in codimension three

Bryant, J. L.

doi:10.1090/S0002-9947-1972-0307245-7

Approximating embeddings of polyhedra in codimension three
HTML articles powered by AMS MathViewer

by J. L. Bryant

Trans. Amer. Math. Soc. 170 (1972), 85-95

DOI: https://doi.org/10.1090/S0002-9947-1972-0307245-7

PDF | Request permission

Abstract:

Let $P$ be a $p$-dimensional polyhedron and let $Q$ be a PL $q$-manifold without boundary. (Neither is necessarily compact.) The purpose of this paper is to prove that, if $q - p \geqslant 3$, then any topological embedding of $P$ into $Q$ can be pointwise approximated by PL embeddings. The proof of this theorem uses the analogous result for embeddings of one PL manifold into another obtained by Černavskiĭ and Miller.

References

R. H. Bing, Radial engulfing, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967) Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 1–18. MR 0238284

Piecewise linear approximations of homeomorphisms

J. L. Bryant and C. L. Seebeck III, Locally nice embeddings of polyhedra, Quart. J. Math. Oxford Ser. (2) 19 (1968), 257–274. MR 234434, DOI 10.1093/qmath/19.1.257
J. C. Cantrell, $n$-frames in euclidean $k$-space, Proc. Amer. Math. Soc. 15 (1964), 574–578. MR 164330, DOI 10.1090/S0002-9939-1964-0164330-1
A. V. Černavskiĭ, Topological embeddings of manifolds, Dokl. Akad. Nauk SSSR 187 (1969), 1247–1250 (Russian). MR 0259918
A. V. Černavskiĭ, Piece-wise linear approximation of embeddings of cells and spheres in codimensions greater than two, Mat. Sb. (N.S.) 80 (122) (1969), 339–364 (Russian). MR 0251730

Taming almost PL embeddings

15

Unknotting close embeddings of polyhedra in codimension greater than two

Robert D. Edwards, The equivalence of close piecewise linear embeddings, General Topology and Appl. 5 (1975), 147–180. MR 370603

Approximating codimension three embeddings

C. L. Seebeck III, Collaring and $(n-1)$-manifold in an $n$-manifold, Trans. Amer. Math. Soc. 148 (1970), 63–68. MR 258045, DOI 10.1090/S0002-9947-1970-0258045-6
Anne Scott, Infinite regular neighborhoods, J. London Math. Soc. 42 (1967), 245–253. MR 212807, DOI 10.1112/jlms/s1-42.1.245
C. Weber, Plongements de polyhèdres dans le domaine métastable, Comment. Math. Helv. 42 (1967), 1–27 (French). MR 238330, DOI 10.1007/BF02564408
Perrin Wright, Radial engulfing in codimension three, Duke Math. J. 38 (1971), 295–298. MR 281214
E. C. Zeeman, Unknotting combinatorial balls, Ann. of Math. (2) 78 (1963), 501–526. MR 160218, DOI 10.2307/1970538

Seminar on combinatorial topology