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Partially acyclic manifold decompositions yielding generalized manifolds. (English) Zbl 0726.57017

Summary: Let G be an upper semicontinuous decomposition (uscd) of the \((n+k)\)- manifold M into subcontinua having the shape of closed orientable n- manifolds \((2<n,k)\). We define G to be j-acyclic if for every element g of G the reduced Čech homology of g vanishes up through dimension j. The primary objective of this investigation is to determine the local connectivity properties of the decomposition space \(B=M/G\) if G is (k-2)- acyclic and B is finite dimensional. The Leray-Grothendieck spectral sequence of the decomposition map p is analyzed, which relegates the principal part of the investigation to studying the structure of the Leray sheaf of p and its relation to the local cohomology of B. Let E denote the subset of B over which the Leray sheaf is not locally constant, K the subset of E over which the Leray sheaf is not locally Hausdorff, and \(D=E-K\). Then we get as our main result, which extends work of R. J. Daverman and J. J. Walsh, and generalizes a result of D. S. Coram and P. Duvall as well:
Theorem. Let G be a (k-2)-acyclic decomposition of the \((n+k)\)-manifold M such that \(k<n+2\), \(B=M/G\) is finite dimensional, and the set E does not locally separate B. Then B is a generalized k-manifold, if either \(k=n+1\), or \(k<n+1\) and M is orientable.

MSC:

57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
55M25 Degree, winding number
55N30 Sheaf cohomology in algebraic topology
55T99 Spectral sequences in algebraic topology
57P99 Generalized manifolds
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[1] F. Ancel, The locally flat approximation of cell-like embedding relations, Ph.D. thesis, University of Wisconsin, Madison, 1976. · Zbl 0405.57007
[2] F. Bonahon and L. Siebenmann, The classification of Seifert fibred 3-orbifolds, Low-dimensional topology (Chelwood Gate, 1982) London Math. Soc. Lecture Note Ser., vol. 95, Cambridge Univ. Press, Cambridge, 1985, pp. 19 – 85. · Zbl 0571.57011 · doi:10.1017/CBO9780511662744.002
[3] Glen E. Bredon, Sheaf theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. · Zbl 0158.20505
[4] Glen E. Bredon, Generalized manifolds, revisited, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 461 – 469.
[5] J. W. Cannon, \Sigma &sup2;\?&sup3;=\?\(^{5}\)/\?, Rocky Mountain J. Math. 8 (1978), no. 3, 527 – 532. · Zbl 0395.57006 · doi:10.1216/RMJ-1978-8-3-527
[6] D. S. Coram and P. F. Duvall Jr., Approximate fibrations, Rocky Mountain J. Math. 7 (1977), no. 2, 275 – 288. · Zbl 0367.55019 · doi:10.1216/RMJ-1977-7-2-275
[7] -, Non-degenerate \( k\)-sphere mappings, Topology Proc. 4 (1979), 67-82. · Zbl 0464.54043
[8] D. S. Coram and P. F. Duvall Jr., Mappings from \?&sup3; to \?&sup2; whose point inverses have the shape of a circle, General Topology Appl. 10 (1979), no. 3, 239 – 246. · Zbl 0417.54014
[9] D. S. Coram and P. F. Duvall Jr., Finiteness theorems for approximate fibrations, Trans. Amer. Math. Soc. 269 (1982), no. 2, 383 – 394. · Zbl 0489.55013
[10] R. J. Daverman, Decompositions of manifolds into codimension one submanifolds, Compositio Math. 55 (1985), no. 2, 185 – 207. · Zbl 0593.57005
[11] R. J. Daverman, The 3-dimensionality of certain codimension-3 decompositions, Proc. Amer. Math. Soc. 96 (1986), no. 1, 175 – 179. · Zbl 0589.54015
[12] Robert J. Daverman, Decompositions of manifolds, Pure and Applied Mathematics, vol. 124, Academic Press, Inc., Orlando, FL, 1986. · Zbl 0608.57002
[13] R. J. Daverman, Decompositions into submanifolds of fixed codimension, Geometric and algebraic topology, Banach Center Publ., vol. 18, PWN, Warsaw, 1986, pp. 109 – 116.
[14] R. J. Daverman and L. S. Husch, Decompositions and approximate fibrations, Michigan Math. J. 31 (1984), no. 2, 197 – 214. · Zbl 0584.57011 · doi:10.1307/mmj/1029003024
[15] Robert J. Daverman and John J. Walsh, A ghastly generalized \?-manifold, Illinois J. Math. 25 (1981), no. 4, 555 – 576. · Zbl 0478.57014
[16] R. J. Daverman and J. J. Walsh, Decompositions into codimension two spheres and approximate fibrations, Topology Appl. 19 (1985), no. 2, 103 – 121. · Zbl 0589.57012 · doi:10.1016/0166-8641(85)90064-1
[17] R. J. Daverman and J. J. Walsh, Decompositions into codimension-two manifolds, Trans. Amer. Math. Soc. 288 (1985), no. 1, 273 – 291. · Zbl 0568.57013
[18] R. J. Daverman and J. J. Walsh, Decompositions into submanifolds that yield generalized manifolds, Topology Appl. 26 (1987), no. 2, 143 – 162. · Zbl 0617.57009 · doi:10.1016/0166-8641(87)90065-4
[19] Jerzy Dydak, On \?\?\(^{n}\)-divisors, Proceedings of the 1978 Topology Conference (Univ. Oklahoma, Norman, Okla., 1978), II, 1978, pp. 319 – 333 (1979). · Zbl 0411.54018
[20] Jerzy Dydak and Jack Segal, Shape theory, Lecture Notes in Mathematics, vol. 688, Springer, Berlin, 1978. An introduction. · Zbl 0401.54028
[21] J. Dydak and J. Segal, Local \?-connectivity of decomposition spaces, Topology Appl. 18 (1984), no. 1, 43 – 58. · Zbl 0547.54027 · doi:10.1016/0166-8641(84)90030-0
[22] Jerzy Dydak and John Walsh, Sheaves that are locally constant with applications to homology manifolds, Geometric topology and shape theory (Dubrovnik, 1986) Lecture Notes in Math., vol. 1283, Springer, Berlin, 1987, pp. 65 – 87. · Zbl 0631.57014 · doi:10.1007/BFb0081420
[23] R. D. Edwards, Suspensions of homology spheres, unpublished manuscript.
[24] Marvin J. Greenberg and John R. Harper, Algebraic topology, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. A first course. · Zbl 0498.55001
[25] Heinz Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Math. Ann. 104 (1931), no. 1, 637 – 665 (German). · Zbl 0001.40703 · doi:10.1007/BF01457962
[26] Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. · Zbl 0029.32203
[27] Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York-Berlin, 1980. Reprint of the 1974 original. · Zbl 0442.00002
[28] W. Hurewicz, Homotopie, Homologie, und lokaler Zussamenhang, Fund. Math. 25 (1935), 467-485. · JFM 61.1362.01
[29] D. Husemoller, Fibre bundles, 2nd ed., Springer-Verlag, New York, 1975. a · Zbl 0307.55015
[30] R. C. Lacher, \?-sphere mappings on \?^{2\?+1}, Geometric topology (Proc. Conf., Park City, Utah, 1974) Springer, Berlin, 1975, pp. 332 – 335. Lecture Notes in Math., Vol. 438.
[31] John McCleary, User’s guide to spectral sequences, Mathematics Lecture Series, vol. 12, Publish or Perish, Inc., Wilmington, DE, 1985. · Zbl 0577.55001
[32] Keiô Nagami, Dimension theory, With an appendix by Yukihiro Kodama. Pure and Applied Mathematics, Vol. 37, Academic Press, New York-London, 1970. · Zbl 0224.54060
[33] Jun-iti Nagata, Modern dimension theory, Revised edition, Sigma Series in Pure Mathematics, vol. 2, Heldermann Verlag, Berlin, 1983. · Zbl 0129.38304
[34] Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. · Zbl 0441.18018
[35] Peter Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401 – 487. · Zbl 0561.57001 · doi:10.1112/blms/15.5.401
[36] H. Seifert, Topologie Dreidimensionaler Gefaserter Räume, Acta Math. 60 (1933), no. 1, 147 – 238 (German). · Zbl 0006.08304 · doi:10.1007/BF02398271
[37] Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. · Zbl 0145.43303
[38] R. Swan, Sheaf theory, Univ. of Chicago Press, Chicago, Ill., 1964.
[39] William P. Thurston, The geometry and topology of three-manifolds, lecture notes. · Zbl 0528.57009
[40] Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, vol. 32, American Mathematical Society, New York, N. Y., 1949. · Zbl 0039.39602
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