Orientation sheaf

In the mathematical field of algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf o_X on X such that the stalk of o_X at a point x is the local homology group

${\displaystyle o_{X,x}=\operatorname {H} _{n}(X,X-\{x\})}$

(in the integer coefficients or some other coefficients).

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

${\displaystyle {\mathcal {V}}_{M}=\Omega _{M}^{n}\otimes {\mathcal {o}}_{M}}$

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:

${\displaystyle \textstyle \int _{M}:\Gamma _{c}(M,{\mathcal {V}}_{M})\to \mathbb {R} .}$

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.