A characterization of manifolds
HTML articles powered by AMS MathViewer
- by Louis F. McAuley PDF
- Trans. Amer. Math. Soc. 209 (1975), 101-107 Request permission
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 $f|(X - C),f|{f^{ - 1}}[0,1/2)$, and $f|{f^{ - 1}}(1/2,1]$ 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
- A. V. Černavskiĭ, Local contractibility of the group of homeomorphisms of a manifold. , Dokl. Akad. Nauk SSSR 182 (1968), 510–513 (Russian). MR 0236948
- 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 0281219
- E. Dyer and M.-E. Hamstrom, Completely regular mappings, Fund. Math. 45 (1958), 103–118. MR 92959, DOI 10.4064/fm-45-1-103-118
- Robert D. Edwards and Robion C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63–88. MR 283802, DOI 10.2307/1970753
- Ralph H. Fox, Covering spaces with singularities, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N.J., 1957, pp. 243–257. MR 0123298
- Louis F. McAuley, Some upper semi-continuous decompositions of $E^{3}$ into $E^{3}$, Ann. of Math. (2) 73 (1961), 437–457. MR 126258, DOI 10.2307/1970312
- Louis F. McAuley, 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 0231348
- Louis F. McAuley, A topological Reeb-Milnor-Rosen theorem and characterizations of manifolds, Bull. Amer. Math. Soc. 78 (1972), 82–84. MR 287524, DOI 10.1090/S0002-9904-1972-12866-0
- Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 77107, DOI 10.2307/1969615
- John Milnor, On manifolds homeomorphic to the $7$-sphere, Ann. of Math. (2) 64 (1956), 399–405. MR 82103, DOI 10.2307/1969983
- John W. Milnor, Sommes de variétes différentiables et structures différentiables des sphères, Bull. Soc. Math. France 87 (1959), 439–444 (French). MR 117744, DOI 10.24033/bsmf.1538 —, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N. J., 1963. MR 29 #634.
- Deane Montgomery and Hans Samelson, Fiberings with singularities, Duke Math. J. 13 (1946), 51–56. MR 15794
- William 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 0283772
- Georges Reeb, Sur certaines propriétés topologiques des variétés feuilletées, Publ. Inst. Math. Univ. Strasbourg, vol. 11, Hermann & Cie, Paris, 1952 (French). MR 0055692 R. H. Rosen, A weak form of the star conjecture for manifolds, Notices Amer. Math. Soc. 7 (1960), 380. Abstract #570-28.
- Melvin C. Thornton, Singularly fibered manifolds, Illinois J. Math. 11 (1967), 189–201. MR 210137 S. Stoilow, Leçons sur les principes topologiques de la théorie fonctions analytiques, Gauthier-Villars, Paris, 1938.
- J. H. C. Whitehead, On simply connected, $4$-dimensional polyhedra, Comment. Math. Helv. 22 (1949), 48–92. MR 29171, DOI 10.1007/BF02568048
- J. H. C. Whitehead, Combinatorial homotopy. I, Bull. Amer. Math. Soc. 55 (1949), 213–245. MR 30759, DOI 10.1090/S0002-9904-1949-09175-9
- Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095, DOI 10.1090/coll/028
- L. C. Siebenmann, Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972), 123–136; ibid. 47 (1972), 137–163. MR 319207, DOI 10.1007/BF02566793
- Gordon Thomas Whyburn, Topological analysis, Second, revised edition, Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N.J., 1964. MR 0165476, DOI 10.1515/9781400879335
- Andrew H. Wallace, Differential topology: first steps, Mathematics Monograph Series, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1968. Second printing. MR 0436148
Additional Information
- © Copyright 1975 American Mathematical Society
- 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