Compact sets definable by free 3-manifolds

Row, W. H.

doi:10.1090/S0002-9947-1974-0362314-2

Compact sets definable by free $3$-manifolds
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by W. H. Row PDF

Trans. Amer. Math. Soc. 197 (1974), 225-244 Request permission

Abstract:

Shape conditions are given that force a compactum (i.e., a compact metric space) embedded in the interior of a nonclosed, piecewise-linear 3-manifold to have arbitrarily close, compact, polyhedral neighborhoods each component of which is a 3-manifold with free fundamental group (i.e., to be definable by free 3-manifolds). For compact, connected ANR’s these conditions reduce to the criterion of having a free fundamental group. Additional conditions are given that insure definability by handlebodies or cubes-with-handles. An embedding of Menger’s universal 1-dimensional curve in Euclidean 3-space is shown to have the property that all tame surfaces, separating in 3-space a fixed pair of points, cannot be adjusted (by a small space homeomorphism) to intersect the embedded curve in a 0-dimensional set.

References

Steve Armentrout, Decompostions of $E^{3}$ with a compact $\textrm {O}$-dimensional set of nondegenerate elements, Trans. Amer. Math. Soc. 123 (1966), 165–177. MR 195074, DOI 10.1090/S0002-9947-1966-0195074-4
R. H. Bing, Tame Cantor sets in $E^{3}$, Pacific J. Math. 11 (1961), 435–446. MR 130679, DOI 10.2140/pjm.1961.11.435
K. Borsuk, Theory of shape, Lecture Notes Series, No. 28, Aarhus Universitet, Matematisk Institut, Aarhus, 1971. MR 0293602
H. G. Bothe, Ein eindimensionales Kompaktum im $E^{3}$, das sich nicht lagetreu in die Mengersche Universalkurve einbetten lässt, Fund. Math. 54 (1964), 251–258 (German). MR 169217, DOI 10.4064/fm-54-3-251-258
H. G. Bothe, A tangled $1$-dimensional continuum in $E^{3}$ which has the cube-with-handles property, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 481–486 (English, with Russian summary). MR 314034
Robert Craggs, Canonical neighborhoods for topologically embedded polyhedra, Trans. Amer. Math. Soc. 173 (1972), 465–490. MR 394687, DOI 10.1090/S0002-9947-1972-0394687-7
Wolfgang Haken, Some results on surfaces in $3$-manifolds, Studies in Modern Topology, Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J., 1968, pp. 39–98. MR 0224071

Erratum for

Some results on surfaces in

manifolds

Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
Solomon Lefschetz, On compact spaces, Ann. of Math. (2) 32 (1931), no. 3, 521–538. MR 1503014, DOI 10.2307/1968249
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
Sibe Mardešić and Jack Segal, Equivalence of the Borsuk and the ANR-system approach to shapes, Fund. Math. 72 (1971), no. 1, 61–68. (errata insert). MR 301694, DOI 10.4064/fm-72-1-61-68
Sibe Mardešić and Jack Segal, Shapes of compacta and ANR-systems, Fund. Math. 72 (1971), no. 1, 41–59. MR 298634, DOI 10.4064/fm-72-1-41-59
William S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, Inc., New York, 1967. MR 0211390
D. R. McMillan Jr., A criterion for cellularity in a manifold. II, Trans. Amer. Math. Soc. 126 (1967), 217–224. MR 208583, DOI 10.1090/S0002-9947-1967-0208583-7
D. R. McMillan Jr., Neighborhoods of surfaces in $3$-manifolds, Michigan Math. J. 14 (1967), 161–170. MR 212778
D. R. McMillan Jr., Compact, acyclic subsets of three-manifolds, Michigan Math. J. 16 (1969), 129–136. MR 243501
D. R. McMillan Jr., Acyclicity in three-manifolds, Bull. Amer. Math. Soc. 76 (1970), 942–964. MR 270380, DOI 10.1090/S0002-9904-1970-12510-1
D. R. McMillan Jr., Boundary-preserving mappings of $3$-manifolds, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 161–175. MR 0276985

properties and related topics

Non-planar embeddings of planar sets in

D. R. McMillan Jr. and Harry Row, Tangled embeddings of one-dimensional continua, Proc. Amer. Math. Soc. 22 (1969), 378–385. MR 246267, DOI 10.1090/S0002-9939-1969-0246267-7
J. Milnor, A unique decomposition theorem for $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. MR 142125, DOI 10.2307/2372800
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
John Stallings, On the loop theorem, Ann. of Math. (2) 72 (1960), 12–19. MR 121796, DOI 10.2307/1970146
John Stallings, On fibering certain $3$-manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 95–100. MR 0158375

Seminar on combinatorial topology