Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T11:26:28.666Z Has data issue: false hasContentIssue false
  • English
  • Français

Resolving zero-dimensional singularities in generalized manifolds

Published online by Cambridge University Press:  24 October 2008

J. L. Bryant  and
R. C. Lacher
J. L. Bryant
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306, U.S.A.
R. C. Lacher
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

A generalized m-manifold is a euclidean neighbourhood retract X that is a homol-ogy m-manifold, i.e. for which H*(X, X–x;Z) = H*(Rm, Rm–0;Z) for each point x of X. In a generalized m-manifold X, a pointy is called singular, or a singularity of X, if p has no neighbourhood homeomorphic to Rm. We shall denote the set of singular points of X by S(X).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 3 , May 1978 , pp. 403 - 413
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Ancel, F. D. and Cannon, J. W. The locally flat approximation of cell-like embedding relation. (Preprint.)Google Scholar
(2)Bing, R. H.A homeomorphism between the 3-sphere and the sum of two solid horned spheres. Ann. of Math. (2) 56 (1952), 354–362.CrossRefGoogle Scholar
(3)Bredon, G. E.Sheaf theory (New York: McGraw-Hill, 1967).Google Scholar
(4)Bredon, G. E. Generalized manifolds, revisited. Topology of manifolds, ed. Cantrell, J. C. and Edwards, C. H. Jr, p. 461 (Chicago: Markham, 1970).Google Scholar
(5)Browder, W.Surgery on simply-connected manifolds, Ergebnisse der Mat., Band 65 (New York: Springer-Verlag, 1972).CrossRefGoogle Scholar
(6)Bryant, J. L., Edwards, R. D. and Seebeck, C. L. III., Approximating codimension-one manifolds with 1-LC embeddings. Notices Amer. Math. Soc. 20 (1973), A–28, Abstract 73T–G17.Google Scholar
(7)Bryant, J. L. and Hollingsworth, J. G.Manifold factors that are manifold quotients. Topology 13 (1974), 19–24.CrossRefGoogle Scholar
(8)Cannon, J. W. Taming codimension one generalized submanifolds of Sn. Topology (to appear).Google Scholar
(9)Cannon, J. W.Σ2H3 = S5/G. Rocky Mountain J. Math. (to appear).Google Scholar
(10)Cannon, J. W. Shrinking cell-like decompositions of manifolds. Codimension three (pre-print).Google Scholar
(11)Cohen, M. and Sullivan, D.Mappings with contractible point-inverses. Notices Amer. Math. Soc. 15 (01 1968), 186.Google Scholar
(12)Cohen, M. and Sullivan, D.On the regular neighborhood of a two-sided submanifold. Topology 9 (1970), 141–147.CrossRefGoogle Scholar
(13)Daverman, R. J.Locally nice codimension one manifolds are locally flat. Bull. Amer. Math. Soc. 79 (1973), 410–413.CrossRefGoogle Scholar
(14)Edwards, R. D.The double suspension of a certain homology 3-sphere is S5. Notices Amer. Math. Soc. 22 (1975), A 334, abstract no. 75T–G33.Google Scholar
(15)Edwards, R. D.Images of manifolds under cell-like maps. Notices Amer. Math. Soc. 25 (1978), abstract no. 752–57–10.Google Scholar
(16)Hudson, J. F. P.Piecewise linear topology (New York: Benjamin, 1975).Google Scholar
(17)Kervaire, M. and Milnor, J.Groups of homotopy spheres: I. Ann. of Math. (2) 77 (1963), 504–537.CrossRefGoogle Scholar
(18)Kirby, R. C. and Siebenmann, L.Foundational essays on topological manifolds, smoothings, and triangulations. Ann. of Math. Studies, no. 88 (Princeton, 1977).CrossRefGoogle Scholar
(19)Lacher, R. C.Cell-like mappings and their generalizations. Bull. Amer. Math. Soc. 83 (1977), 495–552.CrossRefGoogle Scholar
(20)McMillan, D. R. Jr., A criterion for cellularity in a manifold. Ann. of Math. (2) 79 (1964), 327–337.CrossRefGoogle Scholar
(21)Milnor, J. W.Whitehead torsion. Bull. Amer. Math. Soc. 72 (1966), 358–426.CrossRefGoogle Scholar
(22)Rourke, C. P. and Sullivan, D. P.On the Kervaire obstruction. Ann. of Math. (2) 94 (1971), 397–413.CrossRefGoogle Scholar
(23)Siebenmann, L. C. The obstruction to finding a boundary for an open manifold of dimension greater than five. Ph.D. thesis, Princeton, 1975 (Univ. Microfilms no. 66–5012, Ann Arbor, Mich.).Google Scholar
(24)Siebenmann, L. C.Approximating cellular maps with homeomorphisms. Topology 11 (1972), 271–294.CrossRefGoogle Scholar
(25)Smith, B. J.Products of decompositions of En. Trans. Amer. Math. Soc. 184 (1973), 31–41.Google Scholar
(26)Spivak, M.Spaces satisfying Poincaré duality. Topology 6 (1967), 77–102.CrossRefGoogle Scholar
(27)Stan'ko, M. A.The embedding of compacta in euclidean space. Mat. Sbornik 83 (125) (1970), 234–255 [= Math. USSR Sbornik 12 (1970), 234–254].CrossRefGoogle Scholar
(28)Wall, C. T. C.Finiteness conditions for cw complexes. Ann. of Math. (2) 81 (1965), 55–64.CrossRefGoogle Scholar
(29)Wall, C. T. C.Surgery on compact manifolds (New York: Academic Press, 1970).Google Scholar
(30)Weber, C.L' élimination des points doubles dans le cas combinatoire. Comment. Math. Helv. 41 (1966), 179–182.CrossRefGoogle Scholar
(31)West, J. E.Compact ANRs have finite type. Bull. Amer. Math. Soc. 81 (1975), 163–165.CrossRefGoogle Scholar
(32)Wilder, R. L.Topology of manifolds. Amer. Math. Soc. Colloq. Publ. 32 (New York, 1949).CrossRefGoogle Scholar
(33)Galewski, D. E. and Stern, R. J.Classification of simplicial triangulations of topological manifolds. (Preprint.)Google Scholar