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A characterization of manifolds


Author: Louis F. McAuley
Journal: Trans. Amer. Math. Soc. 209 (1975), 101-107
MSC: Primary 57A15
DOI: https://doi.org/10.1090/S0002-9947-1975-0391099-X
MathSciNet review: 0391099
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Abstract: The purpose of this paper is (1) to give a proof of one general theorem characterizing certain manifolds and (2) to illustrate a technique which should be useful in proving various theorems analogous to the one proved here.

Theorem. Suppose that $ f:X \Rightarrow [0,1]$, where $ X$ is a compactum, and that $ f$ has the properties:

(1) for $ 0 \leqslant x < 1/2,{f^{ - 1}}(x) = {S^n} \cong {M_0}$,

(2) $ {f^{ - 1}}(1/2) \cong {S^n}$ with a tame (or flat) $ k$-sphere $ {S^k}$ shrunk to a point,

(3) for $ 1/2 < x \leqslant 1,{f^{ - 1}}(x) \cong $ a compact connected $ n$-manifold $ {M_1} \cong {S^{n - (k + 1)}} \times {S^{k + 1}}$ (a spherical modification of $ {M_0}$ of type $ k$), and

(4) there is a continuum $ C$ in $ X$ such that (letting $ {C_x} = {f^{ - 1}}(x) \cap C$) (a) $ 0 \leqslant x < 1/2,{C_x} \cong {S^k}$, (b) $ {C_{1/2}} = \{ p\} $ a point, (c) for $ 1/2 < x \leqslant 1$, and (d) each of $ f\vert(X - C),f\vert{f^{ - 1}}[0,1/2)$, and $ f\vert{f^{ - 1}}(1/2,1]$ is completely regular.

Then $ X$ is homeomorphic to a differentiable $ (n + 1)$-manifold $ M$ whose boundary is the disjoint union of $ {\bar M_0}$ and $ {\bar M_1}$ where $ {M_i} = {\bar M_i},i = 0,1$.


References [Enhancements On Off] (What's this?)

  • [1] A. V. Černavskiĭ, Local contractibility of the homeomorphism group of a manifold, Dokl. Akad. Nauk SSSR 182 (1968), 510-513 = Soviet Math. Dokl. 9 (1968), 1171-1174. MR 38 #5241. MR 0236948 (38:5241)
  • [2] P. T. Church, Differentiable monotone mappings and open mappings, Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Binghamton, Binghamton, N. Y., 1970), State Univ. of New York at Binghamton, Binghamton, N. Y., 1971, pp. 145-183. MR 43 #6938. MR 0281219 (43:6938)
  • [3] E. Dyer and M.-E. Hamstrom, Completely regular mappings, Fund. Math. 45 (1958), 103-118. MR 19, 1187. MR 0092959 (19:1187e)
  • [4] R. D. Edwards and R. C. Kirby, Deformations and spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63-88. MR 44 #1032. MR 0283802 (44:1032)
  • [5] R. H. Fox, Covering spaces with singularities, Algebraic Geometry and Topology. A Sympos. in Honor of S. Lefschetz, Princeton Univ. Press, Princeton, N. J., 1957, pp. 243-257. MR 23 #A629. MR 0123298 (23:A626)
  • [6] L. F. McAuley, Some upper semi-continuous decompositions of $ {E^3}$ into $ {E^3}$, Ann. of Math (2) 73 (1961), 437-457. MR 23 #A3554. MR 0126258 (23:A3554)
  • [7] -, The existence of a complete metric for a special mapping space and some consequences, Topology Seminar (Wisconsin, 1965), Ann. of Math. Studies, no. 60, Princeton Univ. Press, Princeton, N. J., 1966, pp. 135-139. MR 37 #6903. MR 0231348 (37:6903)
  • [8] -, A topological Reeb-Milnor-Rosen theorem, Bull. Amer. Math. Soc. 78 (1972), 82-84. MR 44 #4728. MR 0287524 (44:4728)
  • [9] E. A. Michael, Continuous selections. I, II, III, Ann. of Math. (2) 63 (1956), 361-382; (2) 64 (1956), 562-580; (2) 65 (1957), 357-390. MR 17, 990; 18, 325; 750. MR 0077107 (17:990e)
  • [10] J. W. Milnor, On manifolds homeomorphic to the $ 7$-sphere, Ann. of Math. (2) 64 (1956), 399-405. MR 18, 498. MR 0082103 (18:498d)
  • [11] -, Sommes de variétés différentiables et structures différentiables der sphères, Bull. Soc. France 87 (1959), 439-444. MR 22 #8518. MR 0117744 (22:8518)
  • [12] -, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N. J., 1963. MR 29 #634.
  • [13] D. Montgomery and H. Samelson, Fiberings with singularities, Duke Math. J. 13 (1946), 51-56. MR 7, 471. MR 0015794 (7:471a)
  • [14] W. L. Reddy, Montgomery-Samelson coverings on manifolds, Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Binghamton, Binghamton, N. Y., 1970), State Univ. of New York at Binghamton, Binghamton, N. Y., 1971, pp. 192-198. MR 44 #1002. MR 0283772 (44:1002)
  • [15] G. Reeb, Sur certaines properiétés topologiques des variétés feuilletées, Actualités Sci. Indust., no. 1183, Hermann, Paris, 1952, pp. 91-154. MR 14, 1113. MR 0055692 (14:1113a)
  • [16] R. H. Rosen, A weak form of the star conjecture for manifolds, Notices Amer. Math. Soc. 7 (1960), 380. Abstract #570-28.
  • [17] Melvin C. Thornton, Singularly fibered manifolds, Illinois Math. J. 11 (1967), 189-201. MR 35 #1031. MR 0210137 (35:1031)
  • [18] S. Stoilow, Leçons sur les principes topologiques de la théorie fonctions analytiques, Gauthier-Villars, Paris, 1938.
  • [19] J. H. C. Whitehead, On simply connected, $ 4$-dimensional polyhedra, Comment. Math. Helv. 22 (1949), 48-92. MR 10, 559. MR 0029171 (10:559d)
  • [20] -, Combinatorial homotopy. I, Bull. Amer. Math. Soc. 55 (1949), 213-245. MR 11, 48. MR 0030759 (11:48b)
  • [21] G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publ., vol. 28, Amer. Math. Soc., Providence, R. I., 1942. MR 4, 86. MR 0007095 (4:86b)
  • [22] L. C. Siebenmann, Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972), 123-163. MR 47 #7752. MR 0319207 (47:7752)
  • [23] G. T. Whyburn, Topological analysis, 2nd rev. ed., Princeton Math. Ser., no. 23, Princeton Univ. Press, Princeton, N. J., 1964. MR 29 #2758. MR 0165476 (29:2758)
  • [24] A. H. Wallace, Differential topology: First Steps, Benjamin, New York, 1968. MR 36 #7150. MR 0436148 (55:9098)

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DOI: https://doi.org/10.1090/S0002-9947-1975-0391099-X
Article copyright: © Copyright 1975 American Mathematical Society

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