Free loop spaces and the Cauchy--Frobenius Lemma

TL;DR

This paper extends the classical Cauchy--Frobenius Lemma to a homotopy equivalence of infinity-groupoids, linking group actions, free loop spaces, and double counting principles in a higher categorical context.

Contribution

It introduces a homotopy-theoretic version of Burnside's Lemma, connecting it with free loop spaces and higher groupoid structures, providing a new perspective on classical counting techniques.

Findings

Homotopy equivalence of infinity-groupoids generalizes Burnside's Lemma.

Establishes a connection between free loop spaces and group actions.

Provides a higher categorical framework for counting fixed points.

Abstract

We upgrade the Cauchy--Frobenius Lemma (`Burnside's Lemma') to a homotopy equivalence of $\infty$-groupoids, essentially given by double counting/Fubini in the free loop space of the quotient.