Exact sequences for ordinary differential cohomology

For $\eta \in {\mathrm{\Omega }}^{k-1}(M)$ define $a(\eta )(z)={\int }_z\eta (\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$. [F9] gives $a(\eta )(\mathrm{\partial }c)={\int }_{c}d\eta (\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$, so $a(\eta )$ is a differential character with curvature $d\eta$. The real cochain $\mathrm{Int}(\eta )$ extends $a(\eta )$ from cycles, and [F9] gives $\mathrm{Int}(d\eta )-\delta \mathrm{Int}(\eta )=0$; hence $I(a(\eta ))=0$. If $\eta -{\eta }^{\mathrm{\prime }}$ is closed with integral periods then $a(\eta )=a({\eta }^{\mathrm{\prime }})$. Conversely, if $a(\eta )=0$, then for every smooth integral $k$-simplex $\sigma$, its boundary is a cycle and [F9] gives ${\int }_{\sigma }d\eta ={\int }_{\mathrm{\partial }\sigma }\eta \in \mathbb{Z}$. In a coordinate chart, take oriented affine $k$-simplices of diameter tending to zero based at any point. Their integrals of $d\eta$ are integer-valued, vary continuously with the simplex scale, and tend to zero, so they vanish for all sufficiently small scales. Dividing by their $k$-dimensional volume and taking the limit shows $d\eta$ vanishes at that point. If $k$ exceeds the chart dimension, $d\eta =0$ for degree reasons. Thus $d\eta =0$ everywhere; since $a(\eta )=0$, all its cycle periods are integral, so $\eta \in {\mathrm{\Omega }}_{\mathbb{Z}}^{k-1}(M)$. The converse follows directly from the definition of integral periods. Therefore $\ker a={\mathrm{\Omega }}_{\mathbb{Z}}^{k-1}(M)$ and the left map of the second row is injective.

If $R(h)=0$, the defining boundary identity says $h(\mathrm{\partial }c)=0$, so $h$ factors through $H_{k-1}^{\mathrm{\infty }}(M;\mathbb{Z})$. Since $\mathbb{R}\mathrm{/}\mathbb{Z}$ is divisible, [F5] identifies the corresponding homomorphisms with smooth singular cohomology $H_{\mathrm{\infty }}^{k-1}(M;\mathbb{R}\mathrm{/}\mathbb{Z})$; [F6] identifies this with $H^{k-1}(M;\mathbb{R}\mathrm{/}\mathbb{Z})$. Conversely each such class defines a flat character. Hence $\ker R=\mathrm{im}{i}_1$ and $i_1$ is injective.

The coefficient cochains on the free smooth integral chain groups form a short exact sequence $0\to C^{\ast }(M;\mathbb{Z})\to C^{\ast }(M;\mathbb{R})\to C^{\ast }(M;\mathbb{R}\mathrm{/}\mathbb{Z})\to 0$: injectivity is coefficient inclusion and AC chooses a real lift on each simplex generator. For a cocycle $u$ with values in $\mathbb{R}\mathrm{/}\mathbb{Z}$, choose a real cochain lift $\tilde{u}$; then $\delta \tilde{u}$ is an integral cocycle. Its class is independent of the lift because any two lifts differ by an integral cochain. If this class is zero, write $\delta \tilde{u}=\delta v$ with $v$ integral; then $\tilde{u}-v$ is a real cocycle lifting $u$. Conversely a class lifted from a real cocycle has zero connecting class. This proves that the kernel of the Bockstein $H^{k-1}(M;\mathbb{R}\mathrm{/}\mathbb{Z})\to H^k(M;\mathbb{Z})$ is the image of real cohomology. For exactness at real cohomology, suppose a real cocycle $r$ reduces to a coboundary modulo integers, say $\bar{r}=\delta u$. Lift $u$ to a real cochain $\tilde{u}$; then $r-\delta \tilde{u}$ is an integral cocycle and maps to $[r]$. In degree zero the same conclusion follows directly because an integer-valued locally constant function is an integral zero-cocycle. Thus the kernel of real reduction is exactly the image of integral cohomology. With the characteristic-class convention $I(h)=[\mathrm{Int}(R(h))-\delta \tilde{h}]$, a flat class represented by $u$ has $I(h)=[-\delta \tilde{u}]=-\beta [u]$; the sign does not change its kernel or image.

Let $c\in H^k(M;\mathbb{Z})$ and choose an integral cocycle representative. By [F7], its real image has a closed-form representative $\omega$. Evaluation on smooth cycles shows that $\omega$ has integral periods. As the real integration cochain and $c_{\mathbb{R}}$ represent the same class, choose a real cochain $\tilde{h}$ with $\delta \tilde{h}=\mathrm{Int}(\omega )-c_{\mathbb{R}}$ and put $h(z)=\tilde{h}(z)(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$. Then $h(\mathrm{\partial }b)={\int }_b\omega (\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$ and the definition of $I$ gives $I(h)=c$. Hence $I$ is onto.

The curvature map lands in ${\mathrm{\Omega }}_{\mathbb{Z}}^k(M)$ by [F2]. It is onto by [F3]. Together with step 1.2 this proves exactness of the first row.

Let $h$ satisfy $I(h)=0$. The cochain formula defining $I$ shows that the real singular cohomology class of $\mathrm{Int}(R(h))$ is the realification of $I(h)$, so [F7] gives a form $\eta$ with $d\eta =R(h)$. Then $h-a(\eta )$ is flat and has zero characteristic class. By step 1.3 its flat class is the reduction of a real cohomology class; [F7] represents that class by a closed $(k-1)$-form $\theta$. Thus $h=a(\eta )+a(\theta )=a(\eta +\theta )$, proving $\ker I\subseteq \mathrm{im}a$. The reverse inclusion is step 1.3 and the form-action calculation in step 1.1.

On $M=S^1$ and $k=2$, [F8] gives a nonzero flat class in $H^1(S^1;\mathbb{R}\mathrm{/}\mathbb{Z})$. By the first exact sequence already proved in steps 1.2 and 2.1 it gives a nonzero character with $R=0$. Its characteristic class lies in $H^2(S^1;\mathbb{Z})=0$ by [F8], so it has the same $(I,R)$ pair as the zero character. This proves that the two maps need not determine a character.