Skip to main content

Sheaves that are locally constant with applications to homology manifolds

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.

Keywords

  • Open Subset
  • Spectral Sequence
  • Torsion Module
  • Homology Theory
  • Principal Ideal Domain

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Supported in part by NSF grants

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Armentrout and T. Price, Decompositions into compact sets with UV properties, Trans. Amer. Math. Soc. 141 (1969), 433–442.

    MathSciNet  MATH  Google Scholar 

  2. E. G. Begle, The Vietoris mapping theorem for bicompact spaces, Ann. of Math., (2) 51 (1950), 534–543.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. G. Bredon, Generalized manifolds revisited, Proceedings of the Georgia Conference 1969. Ed. Cantrell, J. C. and Edwards, C. H. (Markham 1970).

    Google Scholar 

  4. _____, Sheaf Theory, McGraw-Hill, New York, 1967.

    MATH  Google Scholar 

  5. _____, Wilder manifolds are locally orientable, Proc. Nat. Acad. Sci. U.S. 63 (1969), 1079–1081.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. J. Bryant, Homogeneous ENR's, preprint.

    Google Scholar 

  7. J. Dydak and J. Segal, Local n-connectivity of decomposition spaces, Topology and its Appl. 18 (1984), 43–58.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. J. Dydak and J. Walsh, Sheaves with finitely generated isomorphic stalks and homology manifolds, to appear in Proc. Amer. Math. Soc.

    Google Scholar 

  9. __________, Cohomological local connectedness of decomposition spaces, in preparation.

    Google Scholar 

  10. __________, Estimates of the dimension of decomposition spaces, in preparation.

    Google Scholar 

  11. J. Dydak, On the shape of decomposition of decomposition spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 23 (1975), 293–298.

    MathSciNet  MATH  Google Scholar 

  12. G. Kozlowski, Factoring certain maps up to homotopy, Proc. Amer. Math. Soc. 21 (1969), 88–92.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. S. Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957), 604–610.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

Dydak, J., Walsh, J. (1987). Sheaves that are locally constant with applications to homology manifolds. In: Mardešić, S., Segal, J. (eds) Geometric Topology and Shape Theory. Lecture Notes in Mathematics, vol 1283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0081420

Download citation

  • DOI: https://doi.org/10.1007/BFb0081420

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18443-0

  • Online ISBN: 978-3-540-47975-8

  • eBook Packages: Springer Book Archive