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 $(X,x)$.
Categories: math.DG