A collaring theorem for codimension one manifolds

Daverman, Robert J.; Tinsley, Fred C.

doi:10.1090/S0002-9939-1994-1205486-8

A collaring theorem for codimension one manifolds
HTML articles powered by AMS MathViewer

by Robert J. Daverman and Fred C. Tinsley PDF

Proc. Amer. Math. Soc. 120 (1994), 969-972 Request permission

Abstract:

The chief result implies that an $n$-manifold $S$ embedded in the interior of an $(n + 1)$-manifold $M$ as a closed, separating subset is locally flatly embedded if the embedding is well behaved in a locally peripheral sense and if $S$ has arbitrarily close neighborhoods $Q$ such that the fundamental groups of appropriate components of $Q\backslash S$ admit a uniform finite upper bound on the number of generators.

References

C. E. Burgess, Criteria for a $2$-sphere in $S^{3}$ to be tame modulo two points, Michigan Math. J. 14 (1967), 321–330. MR 216481
J. C. Cantrell, Almost locally flat embeddings of $S^{n-1}$ in $S^{n}$, Bull. Amer. Math. Soc. 69 (1963), 716–718. MR 154288, DOI 10.1090/S0002-9904-1963-10998-2
A. V. Černavskiĭ, The identity of local flatness and local simple connectedness for imbeddings of $(n-1)$-dimensional into $n$-dimensional manifolds when $n>4$, Mat. Sb. (N.S.) 91(133) (1973), 279–286, 288 (Russian). MR 0334222
R. J. Daverman, Non-homeomorphic approximations of manifolds with surfaces of bounded genus, Duke Math. J. 37 (1970), 619–625. MR 267546
Robert J. Daverman, Locally nice codimension one manifolds are locally flat, Bull. Amer. Math. Soc. 79 (1973), 410–413. MR 321095, DOI 10.1090/S0002-9904-1973-13190-8
Robert J. Daverman, Every crumpled $n$-cube is a closed $n$-cell-complement, Michigan Math. J. 24 (1977), no. 2, 225–241. MR 488066
R. J. Daverman, Decompositions of manifolds into codimension one submanifolds, Compositio Math. 55 (1985), no. 2, 185–207. MR 795714
Robert J. Daverman, Each crumpled $4$-cube is a closed $4$-cell-complement, Topology Appl. 26 (1987), no. 2, 107–113. MR 896866, DOI 10.1016/0166-8641(87)90061-7
R. J. Daverman and F. C. Tinsley, Laminations, finitely generated perfect groups, and acyclic maps, Michigan Math. J. 33 (1986), no. 3, 343–351. MR 856526, DOI 10.1307/mmj/1029003414
R. J. Daverman and F. C. Tinsley, A controlled plus construction for crumpled laminations, Trans. Amer. Math. Soc. 342 (1994), no. 2, 807–826. MR 1182981, DOI 10.1090/S0002-9947-1994-1182981-6
Ralph H. Fox and Emil Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. (2) 49 (1948), 979–990. MR 27512, DOI 10.2307/1969408
Robion C. Kirby, On the set of non-locally flat points of a submanifold of codimension one, Ann. of Math. (2) 88 (1968), 281–290. MR 236900, DOI 10.2307/1970575
William S. Massey, Algebraic topology: an introduction, Graduate Texts in Mathematics, Vol. 56, Springer-Verlag, New York-Heidelberg, 1977. Reprint of the 1967 edition. MR 0448331
Frank Quinn, Ends of maps. III. Dimensions $4$ and $5$, J. Differential Geometry 17 (1982), no. 3, 503–521. MR 679069