Dijkgraaf-Witten invariant in topological $K$-theory

Given a finite group $G$, we define a new invariant of odd-dimensional oriented closed manifolds and call it the KDW invariant. This invariant is a Dijkgraaf–Witten invariant in terms of $K$-theory. In this paper, we compute the invariant of the Brieskorn homology spheres with $G={\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$. We should remark that, in this computational result, the fundamental groups of the Brieskorn homology spheres and ${\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$ are not nilpotent.