Dijkgraaf-Witten invariant in topological $K$-theory

TL;DR

This paper introduces a new topological invariant called the KDW invariant, based on $K$-theory, and computes it for specific 3-manifolds with non-nilpotent fundamental groups, expanding the understanding of topological invariants.

Contribution

The paper defines a novel Dijkgraaf-Witten type invariant in topological $K$-theory and computes it for Brieskorn homology spheres with non-nilpotent fundamental groups.

Findings

Computed the KDW invariant for Brieskorn homology spheres

Demonstrated the invariant applies to non-nilpotent fundamental groups

Extended the class of manifolds with known $K$-theoretic invariants

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.