Generalized manifolds whose nonmanifold set has neighborhoods bounded by tori
Author:
Matthew G. Brin
Journal:
Trans. Amer. Math. Soc. 264 (1981), 539555
MSC:
Primary 57P99; Secondary 57N10
MathSciNet review:
603780
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Abstract: We show that all compact, ANR, generalized manifolds whose nonmanifold set is 0dimensional and has a neighborhood system bounded by tori are celllike images of compact manifolds if and only if the Poincaré conjecture is true. We also discuss to what extent the assumption of the Poincaré conjecture can be replaced by other hypotheses.
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 [1]
 S. Armentrout, Cellular decompositions of manifolds that yield manifolds, Mem. Amer. Math. Soc. no. 107 (1971). MR 0413104 (54:1225)
 [2]
 R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 456483. MR 0087090 (19:300f)
 [3]
 , Necessary and sufficient conditions that a manifold be , Ann. of Math. (2) 68 (1958), 1737. MR 0095471 (20:1973)
 [4]
 K. Borsuk, Theory of retracts, Monografie Mat., Tom. 44, PWN, Warsaw, 1967. MR 0216473 (35:7306)
 [5]
 M. G. Brin, Improving manifold compactifications of open manifolds, Houston J. Math. 4 (1978), 149163. MR 0494116 (58:13046)
 [6]
 J. W. Cannon, The recognition problem: What is a topological manifold?, Bull. Amer. Math. Soc. 84 (1978), 832867. MR 0494113 (58:13043)
 [7]
 J. W. Cannon, J. L. Bryant and R. C. Lacher, The structure of generalized manifolds having nonmanifold set of trivial dimension, Geometric Topology (Proc. 1977 Georgia Topology Conf.), edited by J. C. Cantrell, Academic Press, New York, 1979, pp. 261303. MR 537735 (80h:57026)
 [8]
 C. H. Edwards, Jr., Open manifolds which are simply connected at infinity, Proc. Amer. Math. Soc. 14 (1963), 391395. MR 0150745 (27:732)
 [9]
 W. Haken, Some results on surfaces in manifolds, Studies in Modern Topology, Math. Assoc. Amer., distributed by PrenticeHall, Englewood Cliffs, N. J., 1968, pp. 3998. MR 0224071 (36:7118)
 [10]
 J. M. Kister and D. R. McMillan, Jr., Locally Euclidean factors of which cannot be embedded in , Ann. of Math. (2) 76 (1962), 541546. MR 0144322 (26:1868)
 [11]
 T. Knoblauch, Imbedding deleted manifold neighborhoods in , Illinois J. Math. 18 (1974), 598601. MR 0380801 (52:1698)
 [12]
 R. C. Lacher, Celllike mappings. I, Pacific J. Math. 30 (1969), 717731. MR 0251714 (40:4941)
 [13]
 , Celllike mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977), 495553. MR 0645403 (58:31095)
 [14]
 D. R. McMillan, Jr., Some contractible open manifolds, Trans. Amer. Math. Soc. 102 (1962), 373382. MR 0137105 (25:561)
 [15]
 , Strong homotopy equivalence of manifolds, Bull. Amer. Math. Soc. 73 (1967), 718722. MR 0229243 (37:4817)
 [16]
 , Acyclicity in threemanifolds, Bull. Amer. Math. Soc. 76 (1970), 942964. MR 0270380 (42:5269)
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 C. D. Papakyriakopoulos, On Dehn's lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 126. MR 0090053 (19:761a)
 [18]
 F. Quinn, Resolutions of homology manifolds, Notices Amer. Math. Soc. 26 (1979), A130, Abstract #763579.
 [19]
 F. Raymond, R. L. Wilder's work on generalized manifoldsan appreciation, Algebraic and Geometric Topology (K. C. Millett, Ed.), Lecture Notes in Math., vol. 664, SpringerVerlag, Berlin and New York, 1978, pp. 732. MR 518402 (80h:01023)
 [20]
 R. Sternfeld, A contractible open manifold that embeds in no compact manifold, Ph.D. Thesis, Univ. Wisconsin, Madison, 1977.
 [21]
 J. Vrabec, Submanifolds of acyclic manifolds, Pacific J. Math. 49 (1973), 243263. MR 0350742 (50:3234)
 [22]
 F. Waldhausen, Irreducible manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 5688. MR 0224099 (36:7146)
 [23]
 A. Wright and R. Messer, Embedding open manifolds in compact manifolds, Pacific J. Math. 82 (1979), 163177. MR 549841 (80m:57004)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198106037808
PII:
S 00029947(1981)06037808
Article copyright:
© Copyright 1981
American Mathematical Society
