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Are 2d gauge anomalies determined by genus-one data?

Let $G$ be a (finite, say) group and $\alpha \in \mathrm{H}^3(\mathrm{B}G; \mathrm{U}(1))$ a 3-cohomology class. For each oriented 3-manifold $X^3$ equipped with a $G$-bundle $P : X \to \mathrm{B}G$, ...