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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.
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.
added this pointer on generalization to $\mathbb{Z}/2$-graded cohomology (as appropriate for twists of KR-theory):
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).
Has anyone written about transgression for groupoid cohomology? The example I have in mind is the following: let $G\rightrightarrows X$ be a groupoid (one can assume the space of arrows is finite). Let $\alpha:G^{(2)}\to\mathbb{C}$ be a 2-cocycle. One can consider the simplicial set $\text{hom}(\partial\Delta,G)$ where $\Delta$ is the standard 2-simplex. Does $\alpha$ transgress to a 1-cocycle on $\text{hom}(\partial\Delta,G)$ via the usual pull-push involving mapping spaces?
I believe I know the answer in the case that $G=\text{Pair}(X).$ In this case $\text{hom}(\partial\Delta,G)=\text{Pair}(X\times X\times X),$ where we identify composable triples starting and ending at the same point with $X\times X\times X,$ ie. composable pairs. We get a 1-cocycle on $\text{hom}(\partial\Delta,G)$ by taking $t^*\alpha-s^*\alpha.$ Of course, this cocycle is trivial, but I want to do transgression at the level of cocycles anyway.
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
Ah I see, great. This formulation uses the inertia groupoid, is there one that uses $\text{hom}(\partial\Delta,G)?$
Just to note that $\partial \Delta^2$ 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 $\partial \Delta^2$ then, no, I haven’t seen that spelled out. But it’s guaranteed to work.
At least in my firefox, formula (1) has bad overlapping of symbols in rendering.
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.
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