Cheeger–Simons differential character

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold without boundary and let $k\ge 1$. With the additive normalization $U(1)=\mathbb{R}\mathrm{/}\mathbb{Z}$, a degree-$k$ Cheeger–Simons differential character is a homomorphism $h:Z_{k-1}^{\mathrm{\infty }}(M;\mathbb{Z})\to \mathbb{R}\mathrm{/}\mathbb{Z}$ for which there exists a closed $k$-form $\omega$ with integral periods satisfying $h(\mathrm{\partial }c)={\int }_c\omega (\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$ for every $c\in C_k^{\mathrm{\infty }}(M;\mathbb{Z})$. The structural lemma Differential-character curvature and characteristic class are well-defined proves that $\omega$ is unique; write $R(h)=\omega$. Under full AC, choose a real cochain lift $\tilde{h}$ extending a lift of $h$ from cycles; the characteristic class is $I(h)=[\mathrm{Int}(\omega )-\delta \tilde{h}]\in H^k(M;\mathbb{Z})$, where $\mathrm{Int}(\omega )(c)={\int }_c\omega$ and smooth singular cohomology is identified with ordinary singular cohomology by the smooth-chain comparison. Its independence of the lift is proved in the same structural lemma.

A smooth differential $k$-form

Smooth integral singular chain complex

Singular cohomology with coefficients

Smooth integral chains compute singular homology

Integral of a form over a smooth singular simplex

De rham cochain complex

Singular cohomology ring

The element ${\omega }_h$ is a real-valued smooth form. “Integral periods” means that its integral over every smooth integral $k$-cycle is an integer. The characteristic class records the integral cocycle obtained from any real cochain extension; its well-definedness is established by Differential-character curvature and characteristic class are well-defined. Degree zero is defined separately.