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What does the notation $\times_\chi$ mean in this context?

Is it semidirect product?

Context: $\pi_n(E)\cong \text{coker}\times_\chi\ker$, for a cohomology class $\chi\in H^2(\ker,\text{coker})$.

Paper: http://www.unirioja.es/cu/anromero/AAECC-ACA11.pdf Page 9

Thanks for any help.

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From the nLab:

Definition 5. (central extension associated to group 2-cocycle)

For $[c] \in H^2_{\mathrm{Grp}}(G,A)$ a group cohomology class represented by a cocycle $c \colon G \times G \to A$, define a group $$G \times_c A \in \mathrm{Grp}$$ as follows. The underlying set is the cartesian product $U(G) \times U(A)$ of the underlying sets of $G$ and $A$. The group operation on this is given by $$(g_1,a_1) \cdot (g_2,a_2)≔(g_1 \cdot g_2, a_1+a_2+c(g_1,g_2))$$

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  • $\begingroup$ The main point being that any central extension can be written as such, so there is such a $\chi$ exists. $\endgroup$ – Najib Idrissi Mar 24 '16 at 14:40

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