Cheeger-Chern-Simons classes of representations of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and the spectrum of rational double point singularities

Abstract

Let $L$ be a compact oriented $3$-manifold and $\rho :{\pi }_1(L)\to \mathrm{GL}(n,\mathbb{C})$ a representation. Evaluating the Cheeger-Chern-Simons class ${\hat{c}}_{\rho ,k}\in H^{2k-1}(L;\mathbb{C}/\mathbb{Z})$ 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 ${\hat{c}}_{\rho ,2}([L])$ 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 ${\hat{c}}_{\rho ,2}([L])$ of any representation $\rho$ in terms of $\tilde{\xi }(D)$. Given a topologically trivial representation $\rho :{\pi }_1(L)\to \mathrm{GL}(n,\mathbb{C})$ we construct an element $⟨L,\rho ⟩$ 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 $(X,x)$.