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For determing the quantomorphism 6-group of the M5 WZW term, I need the degree-5 cohomology with real (discrete) coefficients of the geometric realization of the homotopy pullback of the canonical inclusion along the map which is the M2-brane WZW term.
Since geometric realization does not preserve general homotopy pullbacks, it seems hard to determine it. But there is a canonical map from it to the geometric realization of the -principal bundle underlying the M2-brane WZW term, and that realization is a -fibration over the (homotopy type) of the base space(time) .
So I know at least that the degree-5 real cohomology of that -fibration pulls back to the cohomology that I really need. Eventually I need to understand kernel and cokernel of this pullback map. But for the moment I’d just like to understand the degree-5 cohomology of -fibrations over themselves.
So in the relevant Serre spectral sequence the potential differentials that may change the result from being just are
;
;
I suppose. Is there anything useful to be said about these, in general?
The obvious guess is that these maps are given by the cup product with the 4-class of the -bundle. Are they? This must be a standard fact…
Don’t know if it helps but apparently Harada and Kono study the cohomology of
for a simply connected compact Lie group, according to this p. 5.
Thanks! I have chased references a bit now, but I don’t quite see what I need yet.
I suppose that generally the -differential for the Serre spectral sequence of -fibrations is cup product with the twisting -class and that this follows with the argument in Atiyah-Segal 05, in between (4.1) and (4.2).
(Probably it follows with much more direct arguments, too…)
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