The Barratt--Priddy--Quillen theorem via scanning methods

2025/10/15 by Marie-Camille Delarue, Delarue, Marie-Camille · 1 citation
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2510.13564

openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

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 ℝ^∞ and build a scanning map from this model to a space of local images.

Citations

• Stable homology of Higman--Thompson groups via scanning methods
• Homotopy types of topological stacks of categories
• A short exposition of the Madsen-Weiss theorem
• Stable homology of automorphism groups of free groups
• The stable moduli space of Riemann surfaces: Mumford's conjecture

Cited by

• Stable homology of Higman--Thompson groups via scanning methods

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