Dijkgraaf-Witten invariant in topological $K$-theory
Koki Yanagida
TL;DR
This work extends the Dijkgraaf-Witten invariant to a K-theory framework by defining the $K$-theoretic invariant $\mathrm{KDW}_G([M]_K)$ for odd-dimensional oriented manifolds and introducing the computable $\alpha$-KDW via Knapp's map $\psi_G$. It develops a practical computational approach that reduces to induced representations from cyclic subgroups, aided by Chebyshev-polynomial techniques to analyze $\mathrm{Hom}(\pi_1(M), G)$. The authors obtain explicit formulas for lens spaces and Brieskorn (Brieskorn homology) spheres with $G=\mathrm{PSL}_2(\mathbb{F}_p)$, including cases where $p$ divides among the Seifert data $k_i$, and provide a bordism-based strategy to relate Brieskorn computations to lens-space data. These results show how DW-type invariants extend beyond nilpotent groups and connect representation theory, $K$-theory, and 3-manifold topology in a computable framework. The methods yield concrete counts of representations and faithful inductions, offering new tools for topological quantum field theory insights in non-nilpotent settings.
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.