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

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.

💡 Research Summary

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The paper presents a fresh proof of the Barratt–Priddy–Quillen theorem using scanning methods inspired by recent work of Galatius, Kupers, and Randal‑Williams. The classical theorem states that the homology of the symmetric groups stabilises as the number of letters grows, and that the stable homology is isomorphic to the homology of the infinite loop space underlying the sphere spectrum, i.e.
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