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Resolving acyclic images of three-manifolds

  • J. L. Bryant (a1) and R. C. Lacher (a1)
Abstract

Our main result is that a locally contractible Z2-acyclic image of a 3-manifold without boundary is the cell-like image of a 3-manifold without boundary (all mappings being proper). Consequently, such images are generalized 3-manifolds (that is, finite-dimensional and Z-homology 3-manifolds). A refinement of the proof allows the omission of the Z2-acyclicity hypothesis over a zero-dimensional set; an application is the result of D. R. McMillan, Jr., to the effect that a generalized 3-manifold with compact zero-dimensional singular set admits a resolution if and only if a deleted neighbourhood of the singular set embeds in a compact, orientable 3-manifold.

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(2) F. D. Ancel and J. W. Cannon The locally flat approximation of cell-like embedding relations. Ann. of Math. 109 (1979), 6186.

(4) G. E. Bredon Wilder manifolds are locally orientable. Proc. Nat. Acad. Sci. U.S. 63 (1969), 10791081.

(6) J. W. Cannon Shrinking cell-like decompositions of manifolds: co-dimension three. Ann. of Math. (2) 110 (1979), 83112.

(7) J. W. Cannon The recognition problem: What is a topological manifold? Bull. Amer. Math. Soc. 84 (1978), 832866.

(10) G. Kozlowski and J. J. Walsh The cell-like mapping problem, Bull. Amer. Math. Soc. (2) 2 (1980), 315316.

(11) R. C. Lacher Cell-like mappings and their generalizations. Bull. Amer. Math. Soc. 83 (1977), 495552.

(16) D. R. McMillan Jr Acyclicity in three-manifolds. Bull. Amer. Math. Soc. 76 (1970), 942964.

(17) J. Milnor A unique decomposition theorem for 3-manifolds. Amer. J. Math. 84 (1962), 17.

(21) R. C. Lacher Some mapping theorems. Trans. Amer. Math. Soc. 195 (1974), 291303.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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