Chapter

Geometric Topology and Shape Theory

Volume 1283 of the series Lecture Notes in Mathematics pp 65-87

Date:

Sheaves that are locally constant with applications to homology manifolds

  • Jerzy DydakAffiliated withDepartment of Mathematics, University of TennesseeDepartment of Mathematics, University of California
  • , John WalshAffiliated withDepartment of Mathematics, University of TennesseeDepartment of Mathematics, University of California

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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.