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 . {\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.
See also There is also a definition in terms of dualizing complex in Verdier duality; in particular, one can define a relative orientation sheaf using a relative dualizing complex.
References Kashiwara, Masaki; Schapira, Pierre (2002), Sheaves on Manifolds, Berlin: Springer, ISBN 3540518614
External links Two kinds of orientability/orientation for a differentiable manifold