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nLab > Latest Changes: twisted de Rham cohomology

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2020

    added pointer to the original article:

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2020

    added pointer to

    diff, v7, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2020

    added pointer to:

    diff, v8, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2020

    added pointer to:

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2020

    added pointer to:

    diff, v10, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2021

    added references for twists in degree 1 and for twists in degree higher than 3

    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2021

    am adding references (here) on 3-twisted de Rham cohomology generalized to equivariance/orbifolds

    diff, v13, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2021

    started some lines on the definition of equivariant twisted de Rham cohomology according to Tu&Xu and others (here).

    It’s telegraphic for the moment, since first I wrote out some technical preliminaries (here) which made me run out of steam here.

    More later.

    diff, v15, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2021

    I have added (here) some more lead-in words for the equivariant case (but still thin) and then I started to spell out (here) detail/proof for the claims implicit in FHT 07, (3.5).

    (What I don’t see yet is how to prove that the action of π 1\pi_1 on the cohomology of the fiber is trivial, so that their Serre spectral sequence really applies – have left a comment on this here.)

    diff, v17, current

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 28th 2021

    Added:

    A spectral sequence for twisted de Rham cohomology is discussed in

    • Weiping Li, Xiugui Liu, He Wang, On a spectral sequence for twisted cohomologies, arXiv:0911.1417.

    diff, v23, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2021

    Thanks. I have added publication data (here) and cross-linked with Atiyah&Segal05.

    diff, v24, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 9th 2022
    • (edited Jan 9th 2022)

    for when editing functionality is back:

    we should add pointer to

    • Pierre Deligne, Section II.6 in: Equations différentielles à points singuliers réguliers, Lecture Notes in Math 163, Springer 1970 (pdf, publication.ias:355)

    • Pavel I. Etingof, Igor Frenkel, Alexander A Kirillov, Section 7.2 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998)

    for textbook/moograph discussion of 1-twisted de Rham cohomology.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2022

    added pointer to:

    This is the kind of textbook account that I had been looking for. Will add this also to local system, Gauss-Manin connection and elsewhere.

    diff, v27, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 16th 2022
    • (edited Aug 16th 2022)

    The following fact ought to be classical textbook material, but I am still looking for a real reference:

    Given a complex line bundle with a flat connection ∇\nabla on some smooth manifold, the ∇\nabla-twisted de Rham cohomology of X\mathrm{X} is isomorphic to the untwisted but suitably π 1(X)\pi_1(\mathrm{X})-invariant de Rham cohomology of the universal cover.

    For the record, I have made a note with the exact statement and a pedantic proof: pdf.

    But is there any citable reference that makes this explicit?

    I mean a reference with a real proof that one can point people to who don’t already know how this works. (Is this in Deligne 1970, for the holomorphic version? Maybe I should check again.) And I mean in the generality of complex line bundles that are not assumed to be trivial.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 17th 2022
    • (edited Aug 17th 2022)

    I have now added to the entry here a section with the statement of this proposition:

    H •(Ω dR •(X;ℒ),∇)≃H •((Ω dR •(X^;ℂ)) π 1(X),d) H^\bullet \Big( \Omega_{dR}^\bullet \big( \mathrm{X} ;\, \mathcal{L} \big) ,\, \nabla \Big) \;\;\simeq\;\; H^\bullet \bigg( \Big( \Omega^\bullet_{dR} \big( \widehat{\mathrm{X}} ;\, \mathbb{C} \big) \Big)^{\pi_1(\mathrm{X})} ,\, \mathrm{d} \bigg)

    With a pointer to my pdf note for proof.

    (I have checked again in Deligne 1970, but I don’t see it stated there. It must be stated in some standard account on local systems – if anyone has a citable reference, please drop a note.)

    diff, v30, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2022

    added pointer to:

    diff, v34, current

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