The Barratt--Priddy--Quillen theorem via scanning methods
Marie-Camille Delarue
TL;DR
The paper develops a scanning-method proof of the Barratt--Priddy--Quillen theorem by modeling the symmetric groups as a topological category of configurations and time-ordered paths in $\\mathbb{R}^\infty$, then constructing a scanning map to a space of local images. It builds a sequence of spaces $\\Phi_k^N$ that encode local geometric data and shows they satisfy $\\Phi_k^N \simeq \Omega\\Phi_{k+1}^N$, enabling a delooping to $B\\mathcal{C}_\infty \simeq \Omega^\infty S^\infty$, and ultimately a homology equivalence $B\Sigma_\infty \overset{H_*}{\simeq} \Omega^\infty_0 S^\infty$. A key step is identifying $\\Phi_N^N \simeq S^N$, yielding a spectrum $\\Phi$ equivalent to the sphere spectrum, which links the monoid of configuration-classifying spaces to the group-like infinite loop space. The approach generalizes the scanning-delooping paradigm to broader group families, offering a robust framework for future homology-stability proofs via configuration-cobordism perspectives (e.g., Higman–Thompson groups).
Abstract
The homology of the symmetric groups stabilizes, and the Barratt--Priddy--Quillen theorem identifies the stable homology with that of the infinite loop space underlying the sphere spectrum. We formulate a new proof inspired by Galatius, Kupers, and Randal-Williams using scanning methods. We build a topological model for the monoid formed by all the symmetric groups as a category of paths in $\mathbb{R}^\infty$ and build a scanning map from this model to a space of local images.