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nLab > Latest Changes: transgression in group cohomology

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2021

    In this entry I mean to write out a full proof for the transgression formula for (discrete) group cocycles, using just basic homotopy theory and the Eilenberg-Zilber theorem.

    Currently there is an Idea-section and the raw ingredients of the proof. Still need to write connecting text. But have to interrupt for the moment.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2021
    • (edited Jul 12th 2021)

    have now put a bunch of background material in place to justify the sequence of natural isos (here).

    Still need to justify that this is the correct transgression map on abstract grounds, and to spell out in more detail how to chase a cocycle through it.

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2021

    I have now completed the proof (I think) by adding an argument (here, using a new Prop. here) that the concrete composite of maps via Eilenberg-Zilber (here) is indeed equal to the abstractly defined transgression map (here).

    One could go and spell out yet more detail, but it’s maybe too much detail already, and probably I’ll leave it as is, for the time being.

    diff, v6, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2022

    added this pointer on generalization to ℤ/2\mathbb{Z}/2-graded cohomology (as appropriate for twists of KR-theory):

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2022

    made a bunch of little cosmetic adjustments to text and formulas

    diff, v11, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2022
    • (edited Dec 5th 2022)

    I have enhanced the proof of this Prop, describing the evaluation map on nerves of inertia groupoids.

    (The previous version was glossing over some degeneracies. The new version has more detailed diagrams and full detail on the relevant degeneracies.)

    This is the same material that I just added as an example to function complex (as announced there).

    diff, v15, current

    • CommentRowNumber7.
    • CommentAuthorJosh
    • CommentTimeMay 21st 2023
    • (edited May 21st 2023)

    Has anyone written about transgression for groupoid cohomology? The example I have in mind is the following: let G⇉XG\rightrightarrows X be a groupoid (one can assume the space of arrows is finite). Let α:G (2)→ℂ\alpha:G^{(2)}\to\mathbb{C} be a 2-cocycle. One can consider the simplicial set hom(∂Δ,G)\text{hom}(\partial\Delta,G) where Δ\Delta is the standard 2-simplex. Does α\alpha transgress to a 1-cocycle on hom(∂Δ,G)\text{hom}(\partial\Delta,G) via the usual pull-push involving mapping spaces?

    I believe I know the answer in the case that G=Pair(X).G=\text{Pair}(X). In this case hom(∂Δ,G)=Pair(X×X×X),\text{hom}(\partial\Delta,G)=\text{Pair}(X\times X\times X), where we identify composable triples starting and ending at the same point with X×X×X,X\times X\times X, ie. composable pairs. We get a 1-cocycle on hom(∂Δ,G)\text{hom}(\partial\Delta,G) by taking t *α−s *α.t^*\alpha-s^*\alpha. Of course, this cocycle is trivial, but I want to do transgression at the level of cocycles anyway.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2023

    Yes, the interest in the abstract formulation of transgression as in Def. 2.15 in the entry is that it immediately applies to groupoid cohomogy, too, in fact to cohomology of stacks (as long as the coefficients is a discrete abelian group.

    A comment to this extent is in the paragraph below (11) in arXiv:2212.13836

    • CommentRowNumber9.
    • CommentAuthorJosh
    • CommentTimeMay 21st 2023

    Ah I see, great. This formulation uses the inertia groupoid, is there one that uses hom(∂Δ,G)?\text{hom}(\partial\Delta,G)?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2023
    • (edited May 21st 2023)

    Just to note that ∂Δ 2\partial \Delta^2 is, of course, one model for the circle, hence for the delooping groupoid of the integers, up to weak homotopy equivalence of simplicial sets.

    So in as far as we do not worry about fixing strict groupoid models, any of these choices leads to equivalent formulas.

    If your aim is to work with the specific model where transgressed cocycles are expressed in terms of the components obtained by using specifically ∂Δ 2\partial \Delta^2 then, no, I haven’t seen that spelled out. But it’s guaranteed to work.

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeMay 23rd 2023

    At least in my firefox, formula (1) has bad overlapping of symbols in rendering.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2023
    • (edited May 23rd 2023)

    Thanks for the alert.

    I wonder if this might be due to a bug that somehow got newly introduced. Because, the source code looks correct, and I feel pretty sure that this formula was properly displaying before.

    In any case, I have artifically added some more vertically whitespace now to alleviate the problem. But this is a bad hack and probably makes things worse on systems that render properly. Not sure what to do. Let’s keep an eye on this.

    diff, v18, current

1 to 12 of 12

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