 Orientation Sheaf
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Orientation Sheaf

In algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at a point x is

$o_{X,x}=\operatorname {H} _{n}(X,X-\{x\})$ (in the integer coefficients or some other coefficients).

Let $\Omega _{M}^{k}$ be the sheaf of differential k-forms on a manifold M. If n is the dimension of M, then the sheaf

${\mathcal {V}}_{M}=\Omega _{M}^{n}\otimes {\mathcal {o}}_{M}$ is called the sheaf of (smooth) densities on M. The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map:

$\textstyle \int _{M}:\Gamma _{c}(M,{\mathcal {V}}_{M})\to \mathbb {R} .$ If M is oriented; i.e., the orientation sheaf of the tangent bundle of M is literally trivial, then the above reduces to the usual integration of a differential form.