Toric 2-group anomalies via cobordism

2-group symmetries arise in physics when a 0-form symmetry $G^{[0]}$ and 1-form $H^{[1]}$ intertwine, forming generalised group-like structure. Specialising to the case where both are compact, connected, abelian groups (i.e. tori), we analyse anomalies such `toric symmetries' using cobordism classification. As warm up example, use study various 't Hooft (and phases which they dual) Maxwell theory defined on non-spin manifolds. For our main compute 5th spin bordism group of $B|\mathbb{G}|$ $\mathbb{G}$ is any whose parts $\mathrm{U}(1)$, $|\mathbb{G}|$ geometric realisation nerve $\mathbb{G}$. By leveraging variety algebraic methods, show that $\Omega^{\mathrm{Spin}}_5(B|\mathbb{G}|) \cong \mathbb{Z}/m$ $m$ modulus Postnikov class for $\mathbb{G}$, reproduce expected result appear 4d QED. Moving down two dimensions, recap (anomalous) $\mathrm{U}(1)$ global 2d can be enhanced toric symmetry, before showing its associated local anomaly reduces at most an order 2 anomaly, with