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    • CommentRowNumber1.
    • CommentAuthorJon Beardsley
    • CommentTimeSep 26th 2017

    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 MM is there an object, I’ll call it XX for now, such that (some version of) maps from MM to XX classify twists of de Rham cohomology on MM? 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 YPic(R)Y\to Pic(R), where RR is a ring spectrum, giving me twists of RR-homology of YY.

    Apologies if this is very basic, or perhaps nonsensical.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2017
    • (edited Sep 27th 2017)

    Which twists are you thinking of, besides those classified by B(/2)B (\mathbb{Z}/2)? Do you want to include twists by closed 3-forms, which twist not the \mathbb{Z}-graded but the /2\mathbb{Z}/2-graded de Rham complex?

    • CommentRowNumber3.
    • CommentAuthorJon Beardsley
    • CommentTimeSep 27th 2017
    • (edited Sep 27th 2017)

    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?

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 27th 2017
    • (edited Sep 27th 2017)

    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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2017
    • (edited Sep 28th 2017)

    Okay, I see. Right, so as David says, the /2\mathbb{Z}/2-twisting is for the unoriented case (“pseudo-forms”). Consider the sheaf on CartSp of crossed complexes which to test space UU assigns the crossed complex which is the de Rham complex on UU (shifted to some degree nn) with the groupoid B/2B \mathbb{Z}/2 acting on any kk-forms by sending them to their negative. Write B n/(/2)SmoothGrpd=Sh (CartSp)\flat \mathbf{B}^n \mathbb{R}/(\mathbb{Z}/2) \in Smooth \infty Grpd = Sh_\infty(CartSp) for the image of this under the Dold-Kan correspondene for crossed complexes. By construction this comes with a canonical forgetful morphism

    B n/(/2) B/2 \array{ \flat \mathbf{B}^n \mathbb{R}/(\mathbb{Z}/2) \\ \downarrow \\ B \mathbb{Z}/2 }

    This morphism, regarded as an object in the slice \infty-topos over B/2B \mathbb{Z}/2, is the coefficients for /2\mathbb{Z}/2-twisted de Rham cohomology in degree nn. (We could instead use sheaves of spectra on CartSpaceCartSpace 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

    (dh 1=0) (dh 1=0 dω n=h 1ω n) h 1 h 1 \array{ \left( d h_1 = 0 \right) &\longrightarrow& \left( \array{ d h_1 = 0 \\ d \omega_n = h_1 \wedge \omega_n } \right) \\ h_1 &\mapsto& h_1 }

    Under Lie integration this again yields a morphism in Sh (CartSp)Sh_\infty(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.