A method for shrinking decompositions of certain manifolds
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- by Robert D. Edwards and Leslie C. Glaser
- Trans. Amer. Math. Soc. 165 (1972), 45-56
- DOI: https://doi.org/10.1090/S0002-9947-1972-0295357-6
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Abstract:
A general problem in the theory of decompositions of topological manifolds is to find sufficient conditions for the associated decomposition space to be a manifold. In this paper we examine a certain class of decompositions and show that the nondegenerate elements in any one of these decompositions can be shrunk to points via a pseudo-isotopy. It follows then that the decomposition space is a manifold homeomorphic to the original one. As corollaries we obtain some results about suspensions of homotopy cells and spheres, including a new proof that the double suspension of a Poincaré 3-sphere is a real topological 5-sphere.References
- J. J. Andrews and M. L. Curtis, $n$-space modulo an arc, Ann. of Math. (2) 75 (1962), 1–7. MR 139153, DOI 10.2307/1970414
- R. H. Bing, The cartesian product of a certain nonmanifold and a line is $E^{4}$, Ann. of Math. (2) 70 (1959), 399–412. MR 107228, DOI 10.2307/1970322 —, Decompositions of ${E^3}$, Topology of 3-Manifolds and Related Topics (Proc. Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 5-21. MR 25 #4501.
- R. H. Bing, Radial engulfing, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967) Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 1–18. MR 0238284 J. L. Bryant, Euclidean n-space modulo an $(n - 1)$-cell (to appear).
- E. H. Connell, A topological $H$-cobordism theorem for $n\geq 5$, Illinois J. Math. 11 (1967), 300–309. MR 212808
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- Leslie C. Glaser, On the double suspension of certain homotopy $3$-spheres, Ann. of Math. (2) 85 (1967), 494–507. MR 217797, DOI 10.2307/1970355
- Leslie C. Glaser, On suspensions of homology spheres, Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 8–16. MR 0283804
- Leslie C. Glaser, A decomposition proof that the double suspension of a homotopy $3$-cell is a topological $5$-cell, Illinois J. Math. 16 (1972), 475–490. MR 303541 L. C. Glaser and J. Hollingsworth, Geometrical techniques for studying simplicial triangulations of topological manifolds (to appear).
- Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977 R. C. Kirby and L. C. Siebenmann, For manifolds the Hauptvermutung and the triangulation conjecture are false, Notices Amer. Math. Soc. 16 (1969), 695. Abstract #69T-G80.
- C. Lacher, Some conditions for manifolds to be locally flat, Trans. Amer. Math. Soc. 126 (1967), 119–130. MR 205241, DOI 10.1090/S0002-9947-1967-0205241-X
- 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 —, Upper semicontinuous decompositions of ${E^3}$ into ${E^3}$ and generalizations to metric spaces, Topology of 3-Manifolds and Related Topics (Proc. Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 21-26. MR 25 #4502.
- Ernest Michael, Local properties of topological spaces, Duke Math. J. 21 (1954), 163–171. MR 62424
- Ronald H. Rosen, Concerning suspension spheres, Proc. Amer. Math. Soc. 23 (1969), 225–231. MR 267591, DOI 10.1090/S0002-9939-1969-0267591-8
- L. C. Siebenmann, On detecting Euclidean space homotopically among topological manifolds, Invent. Math. 6 (1968), 245–261. MR 238325, DOI 10.1007/BF01404826 —, A renontriangulable manifolds triangulable ?, Proc. Georgia Conference, 1969, Topology of Manifolds, Markham, Chicago, 1969, pp. 77-84.
- L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271–294. MR 295365, DOI 10.1016/0040-9383(72)90014-6
- John Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481–488. MR 149457
- A. H. Stone, Metrizability of decomposition spaces, Proc. Amer. Math. Soc. 7 (1956), 690–700. MR 87078, DOI 10.1090/S0002-9939-1956-0087078-6
- Perrin Wright, Radial engulfing in codimension three, Duke Math. J. 38 (1971), 295–298. MR 281214
- William L. Voxman, On the shrinkability of decompositions of $3$-manifolds, Trans. Amer. Math. Soc. 150 (1970), 27–39. MR 261577, DOI 10.1090/S0002-9947-1970-0261577-8
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 165 (1972), 45-56
- MSC: Primary 57A30
- DOI: https://doi.org/10.1090/S0002-9947-1972-0295357-6
- MathSciNet review: 0295357