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
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S. Armentrout and T. Price, Decompositions into compact sets with UV properties, Trans. Amer. Math. Soc. 141 (1969), 433–442.
E. G. Begle, The Vietoris mapping theorem for bicompact spaces, Ann. of Math., (2) 51 (1950), 534–543.
G. Bredon, Generalized manifolds revisited, Proceedings of the Georgia Conference 1969. Ed. Cantrell, J. C. and Edwards, C. H. (Markham 1970).
_____, Sheaf Theory, McGraw-Hill, New York, 1967.
_____, Wilder manifolds are locally orientable, Proc. Nat. Acad. Sci. U.S. 63 (1969), 1079–1081.
J. Bryant, Homogeneous ENR's, preprint.
J. Dydak and J. Segal, Local n-connectivity of decomposition spaces, Topology and its Appl. 18 (1984), 43–58.
J. Dydak and J. Walsh, Sheaves with finitely generated isomorphic stalks and homology manifolds, to appear in Proc. Amer. Math. Soc.
__________, Cohomological local connectedness of decomposition spaces, in preparation.
__________, Estimates of the dimension of decomposition spaces, in preparation.
J. Dydak, On the shape of decomposition of decomposition spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 23 (1975), 293–298.
G. Kozlowski, Factoring certain maps up to homotopy, Proc. Amer. Math. Soc. 21 (1969), 88–92.
S. Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957), 604–610.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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
