  Let $L$ be a compact oriented $3$-manifold and $\rho\colon\pi_1(L)\to
\mathrm{GL}(n,\mathbb{C})$ a representation. Evaluating the
Cheeger-Chern-Simons class $\widehat{c}_{\rho,k}\in
H^{2k-1}(L;\mathbb{C}/\mathbb{Z})$ of $\rho$ at $\nu\in H_{2k-1}(L;\mathbb{Z})$
we get characteristic numbers that we call the $k$-th CCS-numbers of $\rho$. We
prove that if $\rho$ is a topologically trivial representation, the 2-nd
CCS-number $\widehat{c}_{\rho,2}([L])$ of the fundamental class $[L]$ of $L$ is
given by the invariant $\tilde{\xi}_\rho(D)$ of the Dirac operator $D$ of $L$
twisted by $\rho$ defined by Atiyah, Patodi and Singer. If $L$ is a rational
homology sphere, we also give a formula for $\widehat{c}_{\rho,2}([L])$ of any
representation $\rho$ in terms of $\tilde{\xi}(D)$. Given a topologically
trivial representation $\rho\colon\pi_1(L)\to\mathrm{GL}(n,\mathbb{C})$ we
construct an element $\langle L,\rho\rangle$ in the $3$-rd algebraic K-theory
group $K_3(\mathbb{C})$ of the complex numbers. For a finite subgroup $\Gamma$
of $\mathrm{SU(2)}$ and its irreducible representations, we compute the 1-st
and 2-nd CCS-numbers. With this, we recover the spectrum of all rational double
point singularities. Motivated by this result, we define the topological
spectrum of rational surface singularities and Gorenstein singularities. Given
a normal surface singularity $(X,x)$ with link a rational homology sphere $L$,
we show how to compute the invariant $\tilde{\xi}_\rho(D)$ for the Dirac
operator of $L$ using either a resolution or a smoothing of $(X,x)$.
