Sheaves that are locally constant with applications to homology manifolds

  • Jerzy Dydak
  • John Walsh
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1283)

Abstract

Analyses are made that establish a connection between properties of presheaves and the constancy of the induced (or associated) sheaves. While the analyses applies regardiess of the source of the presheaves, the applications involve either the homology presheaf and sheaf of a space or the cohomology presheaf and sheaf of a continuous function. Amongst the applications is an elementary proof that homology manifolds are locally orientable; that is, the orientation sheaf is locally constant. Additional applications appearing elsewhere include determining the homological local connectivity of decomposition spaces and providing dimension estimates of the images of closed mappings.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AP]
    S. Armentrout and T. Price, Decompositions into compact sets with UV properties, Trans. Amer. Math. Soc. 141 (1969), 433–442.MathSciNetMATHGoogle Scholar
  2. [Be]
    E. G. Begle, The Vietoris mapping theorem for bicompact spaces, Ann. of Math., (2) 51 (1950), 534–543.MathSciNetCrossRefMATHGoogle Scholar
  3. [Bre1]
    G. Bredon, Generalized manifolds revisited, Proceedings of the Georgia Conference 1969. Ed. Cantrell, J. C. and Edwards, C. H. (Markham 1970).Google Scholar
  4. [Bre2]
    _____, Sheaf Theory, McGraw-Hill, New York, 1967.MATHGoogle Scholar
  5. [Bre3]
    _____, Wilder manifolds are locally orientable, Proc. Nat. Acad. Sci. U.S. 63 (1969), 1079–1081.MathSciNetCrossRefMATHGoogle Scholar
  6. [Bry]
    J. Bryant, Homogeneous ENR's, preprint.Google Scholar
  7. [DS]
    J. Dydak and J. Segal, Local n-connectivity of decomposition spaces, Topology and its Appl. 18 (1984), 43–58.MathSciNetCrossRefMATHGoogle Scholar
  8. [DW1]
    J. Dydak and J. Walsh, Sheaves with finitely generated isomorphic stalks and homology manifolds, to appear in Proc. Amer. Math. Soc.Google Scholar
  9. [DW2]
    __________, Cohomological local connectedness of decomposition spaces, in preparation.Google Scholar
  10. [DW3]
    __________, Estimates of the dimension of decomposition spaces, in preparation.Google Scholar
  11. [Dy]
    J. Dydak, On the shape of decomposition of decomposition spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 23 (1975), 293–298.MathSciNetMATHGoogle Scholar
  12. [Ko]
    G. Kozlowski, Factoring certain maps up to homotopy, Proc. Amer. Math. Soc. 21 (1969), 88–92.MathSciNetCrossRefMATHGoogle Scholar
  13. [Sm]
    S. Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957), 604–610.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Jerzy Dydak
    • 1
    • 2
  • John Walsh
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of TennesseeKnoxville
  2. 2.Department of MathematicsUniversity of CaliforniaRiverside

Personalised recommendations