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A characterization of sheaf-trivial, proper maps with cohomologically locally connected images

Authors
Journal
Topology and its Applications
0166-8641
Publisher
Elsevier
Publication Date
Volume
60
Issue
1
Identifiers
DOI: 10.1016/0166-8641(94)00008-5
Keywords
  • Cohomological Local Connectedness
  • Leray Sheaf
  • Spectral Sequence
  • Shape Fibration

Abstract

Abstract Let f: X → Y be a proper surjection of locally compact metric spaces. Throughout, the Leray sheafs of f are assumed to be (locally) trivial either in all dimensions or through a given dimension. Using a spectral sequence, the cohomological local connectivity of Y is analyzed and thus characterized by the structure of f. We define f to be cohomologically locally connected if, for each y ϵ Y, neighborhood U of y and q ⩾ 0, there is a neighborhood V ⊂ U of y such that the image of the inclusion-induced homomorphism H q ( f −1( U)) → H q ( f −1( V)) is finitely generated. The main result is: Theorem. If f : X → Y is a proper surjection of locally compact metric spaces and each Leray sheaf H q[f] of f is locally constant, then any two of the following statements imply the third : 1. (1) Y is cohomologically locally connected. 2. (2) The stalk of H q[f] is finitely generated, for all q ⩾ 0. 3. (3) f is cohomologically locally connected.

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