Integral-period forms lift to differential characters

Integration of $\omega$ defines a real cochain $w(c)={\int }_c\omega$. [F5] gives $\delta w(c)=w(\mathrm{\partial }c)={\int }_{c}d\omega =0$, so $w$ is a cocycle. By the integral-period hypothesis, evaluation of $[w]$ on $H_k^{\mathrm{\infty }}(M;\mathbb{Z})$ has values in $\mathbb{Z}$; denote this homomorphism by $\lambda$.

Apply [F3] to the free complex $C_{\ast }^{\mathrm{\infty }}(M;\mathbb{Z})$ and coefficient group $\mathbb{Z}$. Its evaluation map onto $\mathrm{Hom}(H_k^{\mathrm{\infty }}(M;\mathbb{Z}),\mathbb{Z})$ is surjective, so choose an integral cocycle $c$ whose evaluation equals $\lambda$.

Under coefficient inclusion $\mathbb{Z}\hookrightarrow \mathbb{R}$, $c$ and $w$ have the same evaluation on $H_k^{\mathrm{\infty }}(M;\mathbb{Z})$. By [F4], evaluation is injective for real coefficients, so $[w-c_{\mathbb{R}}]=0$. Hence there is a real cochain $\tilde{h}\in C_{\mathrm{\infty }}^{k-1}(M;\mathbb{R}):={\mathrm{Hom}}_{\mathbb{Z}}(C_{k-1}^{\mathrm{\infty }}(M;\mathbb{Z}),\mathbb{R})$ with $\delta \tilde{h}=w-c_{\mathbb{R}}$.

Define $h(z)=\tilde{h}(z)(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$ for $z\in Z_{k-1}^{\mathrm{\infty }}(M;\mathbb{Z})$. For every smooth integral $k$-chain $b$, $h(\mathrm{\partial }b)=\delta \tilde{h}(b)(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})=w(b)-c(b)(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})={\int }_b\omega (\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$, because $c(b)$ is integral. Thus $h$ is a differential character with curvature $\omega$. The construction uses a choice of cocycle and primitive, but the statement asserts existence only.