Special and general types Special notions Variants Extra structure Operations Theorems Twisted de Rham cohomology is the twisted cohomology-version of de Rham cohomology, a simple example of twisted differential cohomology. For degree-3 twists this is the codomain of the twisted Chern character on twisted K-theory, and in its orbifold cohomology-generalization it is the codomain of the twisted equivariant Chern character on twisted equivariant K-theory. (1-twisted de Rham cohomology) (…) (3-twisted de Rham cohomology) For a smooth manifold and a closed differential 3-form, the twisted de Rham complex is the -graded vector space equipped with the -twisted de Rham differential Notice that this is nilpotent, due to the odd degree of , such that , and the closure of , . There is also the cohomology of the chain complex whose maps are just multiplication by . This is also called H-cohomology (Cavalcanti 03, p. 19). We discuss notions of twisted de Rham cohomology on (global quotient) orbifolds, as they are used for the codomain of the twisted equivariant Chern character on twisted equivariant K-theory. This combines the above twistings in degrees 1 and 3, the latter induced from the curvature 3-form on a twisting 3-class, the former (an “inner local system”) induced from the flat connection on the principal U(1)-bundle (over the inertia orbifold) which is classified by the transgression of the 3-twist to a 2-class. This requires that this connection be flat, hence that the transgressed 2-class is torsion, which is guaranteed by a Lemma that we discuss first, in Transgression of the 3-twist to a 1-twist on Inertia. In all of the the following: Let be a finite group. For any element, write for its centralizer subgroup. Write for the direct product group with the additive abelian group of integers. Let be a proper smooth G-manifold, hence a smooth manifold equipped with a proper action (from the right, say) of by diffeomorphisms. Notice that for any the fixed...