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Published in:
2016 | OriginalPaper | Chapter
11. Cohomology of Constant Sheaves
Author : Torsten Wedhorn
Published in: Manifolds, Sheaves, and Cohomology
Publisher: Springer Fachmedien Wiesbaden
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Abstract
In this chapter we focus on cohomology of constant sheaves. We start by defining singular (co)homology and then show that for a locally contractible space X singular cohomology with values in a ring R and the cohomology of the constant sheaf R
X
are equal. We deduce that R
X
is acyclic if X is contractible and locally contractible.
Combining this result for \(X=[0,1]\) and the proper base change theorem we deduce homotopy invariance of cohomology of constant sheaves for continuous maps between arbitrary topological spaces in Sect. 11.3. We conclude the chapter with some easy applications.
Notation: Let R always be a commutative ring and let X be a topological space.