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An open collar theorem for $ 4$-manifolds


Author: Craig R. Guilbault
Journal: Trans. Amer. Math. Soc. 331 (1992), 227-245
MSC: Primary 57N13; Secondary 57N40, 57N45
DOI: https://doi.org/10.1090/S0002-9947-1992-1038016-7
MathSciNet review: 1038016
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Abstract: Let $ {M^4}$ be an open $ 4$-manifold with boundary. Conditions are given under which $ {M^4}$ is homeomorphic to $ \partial M \times [0,1)$. Applications include a $ 4$-dimensional weak $ h$-cobordism theorem and a classification of weakly flat embeddings of $ 2$-spheres in $ {S^4}$. Specific examples of $ (n - 2)$-spheres embedded in $ {S^n}$ (including $ n = 4$) are also discussed.


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  • [1] E. Artin, Zur isotopie zweidimensionalen flächen im $ {R_4}$, Abh. Math. Sem. Univ. Hamburg 4 (1926), 174-177.
  • [2] R. H. Bing, A surface is tame if its complement is $ 1-ULC$, Trans. Amer. Math. Soc. 101 (1961), 292-305. MR 0131265 (24:A1117)
  • [3] -, Radial engulfing, Conf. Topology Manifolds, 1967, (J. G. Hocking, ed.), Prindle, Weber and Schmidt, Boston, Mass., 1968, pp. 1-18. MR 0238284 (38:6560)
  • [4] W. S. Boyd and A. H. Wright, A $ 1$-alg simple closed curve in $ {E^3}$ is tame, Canad. J. Math. 25 (1973), 646-656. MR 0324705 (48:3055)
  • [5] E. M. Brown and T. W. Tucker, On proper homotopy theory for noncompact $ 3$-manifolds, Trans. Amer. Math. Soc. 188 (1974), 105-126. MR 0334225 (48:12544)
  • [6] J. L. Bryant and C. L. Seebeck III, Locally nice embeddings in codimension three, Quart. J. Math. Oxford Ser. 21 (1980), 265-272. MR 0290376 (44:7560)
  • [7] A. V. Černavskii, The equivalence of local flatness and local $ 1$-connectedness for imbeddings of $ (n - 1)$-dimensional manifolds in $ n$-dimensional manifolds, Mat. Sb. 91 (133) (1973), 279-286; English transl., Math. USSR-Sb. 20 (1973), 297-304. MR 0334222 (48:12541)
  • [8] M. M. Cohen, A course in simple homotopy theory, Graduate Texts in Math., no. 10, Springer-Verlag, Berlin, 1973. MR 0362320 (50:14762)
  • [9] T. A. Chapman, Locally homotopically unknotted embeddings of manifolds in codimension two are locally flat, Topology 18 (1979), 339-348. MR 551015 (81a:57016)
  • [10] R. J. Daverman, On weakly flat $ 1$-spheres, Proc. Amer. Math. Soc. 38 (1973), 207-210. MR 0310894 (46:9992)
  • [11] -, Locally nice codimension one embeddings are locally flat, Bull. Amer. Math. Soc. 79 (1973), 410-413. MR 0321095 (47:9628)
  • [12] -, Homotopy classification of complements of locally flat codimension two spheres, Amer. J. Math. 98 (1976), 367-374. MR 0420632 (54:8645)
  • [13] -, Decompositions of manifolds, Academic Press, 1986. MR 872468 (88a:57001)
  • [14] R. J. Daverman and T. B. Rushing, Weak flatness for codimension $ 2$ spheres in codimension $ 1$ manifolds, General Topology Appl. 6 (1976), 101-115. MR 0394681 (52:15480)
  • [15] P. F. Duvall, Weakly flat spheres, Michigan Math. J. 16 (1969), 117-124. MR 0246300 (39:7604)
  • [16] F. T. Farrell and L. E. Jones, $ K$-theory and dynamics. I, Ann. of Math. (2) 124 (1986), 531-539. MR 866708 (88f:57062)
  • [17] R. H. Fox and E. Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. (2) 49 (1948), 979-990. MR 0027512 (10:317g)
  • [18] M. H. Freedman, The topology of four-manifolds, J. Differential Geom. 17 (1982), 357-453. MR 679066 (84b:57006)
  • [19] -, The disk theorem for four-dimensional manifolds, Proc. Internat. Congr. Math., Warsaw, 1983, pp. 647-663. MR 804721 (86m:57016)
  • [20] M. H. Freedman and F. Quinn, Topology of $ 4$-manifolds, Princeton Univ. Press, 1990. MR 1201584 (94b:57021)
  • [21] G. G. Garza, Weakly flat curves, Topology Proc. 5 (1980), 71-76. MR 624463 (82j:57014)
  • [22] J. P. Hempel and D. R. McMillan, Locally nice embeddings of manifolds, Amer. J. Math. 88 (1976), 1-19. MR 0205257 (34:5090)
  • [23] J. G. Hollingsworth and T. B. Rushing, Homotopy characterizations of weakly flat codimension $ 2$ spheres, Amer. J. Math. 98 (1976), 385-394. MR 0420631 (54:8644)
  • [24] L. S. Husch and T. M. Price, Finding a boundary for a $ 3$-manifold, Ann. of Math. (2) 91 (1970) 223-235. MR 0264678 (41:9269)
  • [25] V. Liem and G. Venema, Homotopy characterization of weakly flat $ 2$-spheres in $ {S^4}$ (to appear).
  • [26] D. R. McMillan, Jr., A criterion for cellularity in a manifold, Ann. of Math. (2) 79 (1964), 327-337. MR 0161320 (28:4528)
  • [27] M. H. A. Newman and J. H. C. Whitehead, On the group of a certain linkage, Quart. J. Math. 8 (1937), 21-41.
  • [28] F. Quinn, Ends of maps. III: dimensions $ 4$ and $ 5$, J. Differential Geom. 17 (1982), 503-521. MR 679069 (84j:57012)
  • [29] W. H. Row and J. J. Walsh, A nonshrinkable decomposition of $ {S^3}$ whose nondegenerate elements are contained in a cellular arc, Trans. Amer. Math. Soc. 289 (1985), 227-252. MR 779062 (86i:57014)
  • [30] L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Ph.D. dissertation, Princeton Univ., Princeton, N.J., 1965.
  • [31] -, On detecting open collars, Trans. Amer. Math. Soc. 142 (1969), 201-227. MR 0246301 (39:7605)
  • [32] -, Infinite simple homotopy types, Indag. Math. 32 (1970), 479-495. MR 0287542 (44:4746)
  • [33] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [34] J. Stallings, The p.l. structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481-488. MR 0149457 (26:6945)
  • [35] G. A. Venema, Neighborhoods of compacta in $ 4$-manifolds, preprint, 1987. MR 984106 (90a:57023)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1038016-7
Keywords: Open collar, $ 4$-manifold, weakly flat, $ 1$-alg, wild embedding, weak $ h$-cobordism theorem
Article copyright: © Copyright 1992 American Mathematical Society

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