Flat circle character with nontrivial holonomy

For a smooth integral $1$-cycle $z$, define $[z]\in \mathbb{Z}$ by its homology class under [F1] and [F3]. Homology is additive, so $z\mapsto [z]$ is a homomorphism and $h_{\theta }$ is a homomorphism. If $c$ is any smooth integral $2$-chain, then $[\mathrm{\partial }c]=0$ in homology; consequently $h_{\theta }(\mathrm{\partial }c)=0={\int }_{c}0(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$. The defining boundary identity holds with curvature $0$.

By [F2] under [A1], $H^2(S^1;\mathbb{Z})=0$, so the characteristic class of every degree-two character on $S^1$, including $h_{\theta }$, is zero. Its curvature was shown to be zero in step 1.1. On the positive fundamental cycle $\gamma$, $[\gamma ]=1$, hence $h_{\theta }(\gamma )=\theta$. If $\theta \mathrm{\ne }{\theta }^{\mathrm{\prime }}$ in $\mathbb{R}\mathrm{/}\mathbb{Z}$, these values differ, so the characters are distinct despite equal $I$ and $R$. This is an abstract mathematical example; it asserts no physical measurement.