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Hi,

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?

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Could someone retag, please? Say, some "sheaf-theory" and "at.algebraic-topology". –  Anton Fonarev Jul 6 '12 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.

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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. –  Yosemite Sam Jul 6 '12 at 15:09
    
what does the trivial orientation sheaf mean? –  zatilokum Aug 1 '12 at 22:59
    
@zatilokum It means that this is a constant sheaf. –  Anton Fonarev Aug 2 '12 at 12:04

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