I have been reading the paper https://arxiv.org/pdf/1902.04050.pdf by Zemke and at some point we have the following:
My question is, how can one make sense of torsion $\mathrm{Spin}^c$ structures on $S^1\times S^2$? I initially though the $\mathrm{spin}^c$ structures are in correspondence with $H^2(S^1\times S^2)\cong\mathbb{Z}$, which has no torsion.