Orientation sheaf
In the mathematical field of 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 the local homology group o X , x = H n ( X , X − { x } ) {\displaystyle o_{X,x}=\operatorname {H} _{n}(X,X-\{x\})} (in the integer coefficients or some other coefficients). Let Ω M k {\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 V M = Ω M n ⊗ o M {\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: ∫ M : Γ c ( M , V M ) → R .