Counting homotopy classes of mappings via Dijkgraaf-Witten invariants

TL;DR

This paper establishes a connection between the counting of homotopy classes of maps with a fixed degree and Dijkgraaf-Witten invariants, revealing that the number of such classes depends only on the degree modulo the group order.

Contribution

It introduces a novel method to count homotopy classes of maps using Dijkgraaf-Witten invariants, linking algebraic topology with quantum invariants.

Findings

The set of homotopy classes with fixed degree is finite.

The cardinality depends only on the degree modulo the group order.

The cardinality can be expressed explicitly in terms of Dijkgraaf-Witten invariants.

Abstract

Suppose $\Gamma$ is a finite group acting freely on $S^{n}$ ($n\geqslant 3$ being odd) and $M$ is any closed oriented $n$-manifold. We show that, given an integer $k$, the set $\deg^{-1}(k)$ of based homotopy classes of mappings with degree $k$ is finite and its cardinality depends only on the congruence class of $k$ modulo $\#\Gamma$; moreover, $\#\deg^{-1}(k)$ can be expressed in terms of the Dijkgraaf-Witten invariants of $M$.