Dijkgraaf-Witten invariant in topological $K$-theory

Abstract

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 $G=\mathrm{PSL}_2(\mathbb{F}_p)$. We should remark that, in this computational result, the fundamental groups of the Brieskorn homology spheres and $\mathrm{PSL}_2(\mathbb{F}_p)$ are not nilpotent.