Orientation sheaf

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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\})} {\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}} {\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}} {\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} .} {\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