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1. • CommentRowNumber1.
   • CommentAuthorJon Beardsley
   • CommentTimeSep 27th 2017
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   Author: Jon Beardsley
   Format: MarkdownItexHi 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 $Y\to Pic(R)$, where $R$ is a ring spectrum, giving me twists of $R$-homology of $Y$. Apologies if this is very basic, or perhaps nonsensical.

   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 $Y\to Pic(R)$, where $R$ is a ring spectrum, giving me twists of $R$-homology of $Y$.

   Apologies if this is very basic, or perhaps nonsensical.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 27th 2017
   • (edited Sep 27th 2017)
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   Author: Urs
   Format: MarkdownItexWhich twists are you thinking of, besides those classified by $B (\mathbb{Z}/2)$? Do you want to include twists by closed 3-forms, which twist not the $\mathbb{Z}$-graded but the $\mathbb{Z}/2$-graded de Rham complex?

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

3. • CommentRowNumber3.
   • CommentAuthorJon Beardsley
   • CommentTimeSep 27th 2017
   • (edited Sep 27th 2017)
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   Author: Jon Beardsley
   Format: MarkdownItexWell 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?

   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?

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 28th 2017
   • (edited Sep 28th 2017)
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   Author: DavidRoberts
   Format: MarkdownItexThe 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 28th 2017
   • (edited Sep 28th 2017)
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   Author: Urs
   Format: MarkdownItexOkay, I see. Right, so as David says, the $\mathbb{Z}/2$-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 $B \mathbb{Z}/2$ acting on any $k$-forms by sending them to their negative. Write $\flat \mathbf{B}^n \mathbb{R}/(\mathbb{Z}/2) \in Smooth \infty Grpd = Sh_\infty(CartSp)$ for the image of this under the [[Dold-Kan correspondene]] for crossed complexes. By construction this comes with a canonical forgetful morphism $$ \array{ \flat \mathbf{B}^n \mathbb{R}/(\mathbb{Z}/2) \\ \downarrow \\ B \mathbb{Z}/2 } $$ This morphism, regarded as an object in the slice $\infty$-topos over $B \mathbb{Z}/2$, is the coefficients for $\mathbb{Z}/2$-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]] $$ \array{ \left( d h_1 = 0 \right) &\longrightarrow& \left( \array{ d h_1 = 0 \\ d \omega_n = h_1 \wedge \omega_n } \right) \\ h_1 &\mapsto& h_1 } $$ Under [[Lie integration]] this again yields a morphism in $Sh_\infty(CartSp)$, 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.

   Okay, I see. Right, so as David says, the $\mathbb{Z}/2$-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 $B \mathbb{Z}/2$ acting on any $k$-forms by sending them to their negative. Write $\flat \mathbf{B}^n \mathbb{R}/(\mathbb{Z}/2) \in Smooth \infty Grpd = Sh_\infty(CartSp)$ for the image of this under the Dold-Kan correspondene for crossed complexes. By construction this comes with a canonical forgetful morphism

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

   This morphism, regarded as an object in the slice $\infty$-topos over $B \mathbb{Z}/2$, is the coefficients for $\mathbb{Z}/2$-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

   $$\array{ \left( d h_1 = 0 \right) &\longrightarrow& \left( \array{ d h_1 = 0 \\ d \omega_n = h_1 \wedge \omega_n } \right) \\ h_1 &\mapsto& h_1 }$$

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