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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 $\Omega^3(-) \to \mathbf{B}^3 U(1)_{conn}$ along the map $X \to \mathbf{B}^3 U(1)_{conn}$ 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 $\mathbf{B}^2 U(1)$-principal bundle underlying the M2-brane WZW term, and that realization is a $K(\mathbb{Z},3)$-fibration over the (homotopy type) of the base space(time) $X$.
So I know at least that the degree-5 real cohomology of that $K(\mathbb{Z},3)$-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 $K(\mathbb{Z},3)$-fibrations over $X$ themselves.
So in the relevant Serre spectral sequence the potential differentials that may change the result from being just $H^5(X) \oplus H^2(X)$ are
$H^1(X)\stackrel{d_4}{\to} H^5(X)$;
$H^2(X) \stackrel{d_4}{\to} H^6(X)$;
I suppose. Is there anything useful to be said about these, in general?
The obvious guess is that these $d_4$ maps are given by the cup product with the 4-class of the $K(\mathbb{Z},3)$-bundle. Are they? This must be a standard fact…
Don’t know if it helps but apparently Harada and Kono study the cohomology of
$K(\mathbb{Z}, 3) \to (B G)\langle 4 \rangle \to B G$for $G$ 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 $d_{n+1}$-differential for the Serre spectral sequence of $K(\mathbb{Z},n)$-fibrations is cup product with the twisting $n$-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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