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

$o_{X,x}={\mathrm{H}}_n(X,X-\{x\})$

(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

${𝒱}_M={\Omega }_M^n\otimes {ℴ}_M$

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:

$\underset{M}{\int }:{\Gamma }_c(M,{𝒱}_M)\to ℝ.$

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.

External links

• Two kinds of orientability/orientation for a differentiable manifold