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Hi all,
I suspect this is basic stuff, and could be sussed out from something that you all have written (perhaps Urs’ big book), but if I have a manifold M is there an object, I’ll call it X for now, such that (some version of) maps from M to X classify twists of de Rham cohomology on M? In other words, the trivial map should just give us regular de Rham cohomology, and non-trivial maps should give us twists. I’m thinking in analogy with a map Y→Pic(R), where R is a ring spectrum, giving me twists of R-homology of Y.
Apologies if this is very basic, or perhaps nonsensical.
Which twists are you thinking of, besides those classified by B(ℤ/2)? Do you want to include twists by closed 3-forms, which twist not the ℤ-graded but the ℤ/2-graded de Rham complex?
Well I don’t really know this stuff very well, I thought you could generally twist the differential of the de Rham complex by a 1-form. I’m not sure where this fits into what you’re saying. I guess it must be the twists classified by B(Z/2) that you mention?
The Z/2 twists would correspond to a two-fold covering space. Presumably one could take a flat connection on that, i.e. some closed 1-form to make some sort of twisting.
Okay, I see. Right, so as David says, the ℤ/2-twisting is for the unoriented case (“pseudo-forms”). Consider the sheaf on CartSp of crossed complexes which to test space U assigns the crossed complex which is the de Rham complex on U (shifted to some degree n) with the groupoid Bℤ/2 acting on any k-forms by sending them to their negative. Write ♭Bnℝ/(ℤ/2)∈Smooth∞Grpd=Sh∞(CartSp) for the image of this under the Dold-Kan correspondene for crossed complexes. By construction this comes with a canonical forgetful morphism
♭Bnℝ/(ℤ/2)↓Bℤ/2This morphism, regarded as an object in the slice ∞-topos over Bℤ/2, is the coefficients for ℤ/2-twisted de Rham cohomology in degree n. (We could instead use sheaves of spectra on CartSpace instead, to unify the varying degrees more elegantly, but this is a minor technical point.)
For the twists by closed differential 1-forms, consider the homomorphism of semifree dgc-algebras
(dh1=0)⟶(dh1=0dωn=h1∧ωn)h1↦h1Under Lie integration this again yields a morphism in Sh∞(CartSp), and, regarded as an object in the slice over its codomain, this is the coefficient for twisted de Rham cohomology with twists by closed 1-forms, I think.
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