I try to understand the orientation sheaves. When searching it in the google, i meet new areas such as local coefficient system and locally constant sheaves. I realize that any system of local coefficients on X is a locally constant sheaves. But what is the relation with orientation sheaves. Which refferences are there to read it?

  • $\begingroup$ Could someone retag, please? Say, some "sheaf-theory" and "at.algebraic-topology". $\endgroup$ Jul 6, 2012 at 11:00

1 Answer 1


These are purely topological notions and have nothing to do with algebraic geometry in particular.

Let $M$ for simplicity be a topological manifold of dimension $n$. Then the orientation sheaf $\mathcal{L}_{or}(M)$ is the sheafification of the presheaf $U\mapsto H_n(M,M-U;\mathbb{Z})$. It's always a locally constant sheaf with stalks equal to $\mathbb{Z}$. One immediately checks that $\mathcal{L}_{or}$ is trivial if and only if $M$ is orientable. This definition can be generalized.

As for the references, I'd suggest checking A.Dimca, Sheaves in Topology or B.Iversen, Cohomology of Sheaves.

  • 1
    $\begingroup$ to be more explicit one could say that the stalks of $L_{or}(M)$ depend only on the presheaf you defined. To see what the stalks are we can reduce to $\mathbb{R}^n$ as $M$ is a manifold. There we can take a direct system of concentric balls. The relative homology then becomes the n-th homology of the sphere $S^n$ which is $\mathbb{Z}$. Hence $L_{or}(M)$ is indeed locally constant. $\endgroup$ Jul 6, 2012 at 15:09
  • $\begingroup$ what does the trivial orientation sheaf mean? $\endgroup$
    – zatilokum
    Aug 1, 2012 at 22:59
  • $\begingroup$ @zatilokum It means that this is a constant sheaf. $\endgroup$ Aug 2, 2012 at 12:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.