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Counting homotopy classes of mappings via Dijkgraaf-Witten invariants

Haimiao Chen

Abstract

Suppose $Γ$ 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 $°^{-1}(k)$ of based homotopy classes of mappings with degree $k$ is finite and its cardinality depends only on the congruence class of $k$ modulo $\#Γ$; moreover, $\#°^{-1}(k)$ can be expressed in terms of the Dijkgraaf-Witten invariants of $M$.

Counting homotopy classes of mappings via Dijkgraaf-Witten invariants

Abstract

Suppose is a finite group acting freely on ( being odd) and is any closed oriented -manifold. We show that, given an integer , the set of based homotopy classes of mappings with degree is finite and its cardinality depends only on the congruence class of modulo ; moreover, can be expressed in terms of the Dijkgraaf-Witten invariants of .

Paper Structure

This paper contains 2 sections, 2 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Mappings to topological spherical space forms

Key Result

Lemma 2.1

Suppose $M$ is an $n$-manifold, and $f_{0},f_{1} \colon M\rightarrow S^{n}/\Gamma$ are two mappings such that $\pi_{1}(f_{0})=\pi_{1}(f_{1}):\pi_{1}(M)\rightarrow\Gamma$, then $\deg f_{0}\equiv\deg f_{1}\pmod{m}$. If furthermore $\deg f_{0}=\deg f_{1}$, then $f_{0}$ is homotopic to $f_{1}$.

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Example 2.4