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

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 \((X,x)\).