On the wreath transfer in the homology of symmetric groups

Dezhou Li

Abstract

For an odd prime $p$, we give an explicit formula for the wreath transfer $\mathrm{tr} \colon H_*(Σ_{p^2};\mathbb{F}_p)\longrightarrow H_*(Σ_p\wrΣ_p;\mathbb{F}_p)$ associated with the standard inclusion $Σ_p\wrΣ_p\leΣ_{p^2}$. The proof is a direct algebraic computation, using Cartan's classical double-coset formula together with a generating-function argument. As an immediate consequence, we obtain an elementary proof of the validity of the mixed Adem relations in Goodwillie calculus.

Disclosure

“s arising in the study of configuration spaces. I would also like to thank Guchuan Li for his support and for helping me extend the formula to the general case. Finally, I thank Dev Sinha for helpful conversations and encouragement. I used ChatGPT for language editing and for assistance with the coefficient-extraction formulation of the swap contribution (Proposition 4.8) in Section 4. All mathematical content was checked by the author. 2. Preliminaries”
PDF page 4

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T18:58:00.933776Z

Structural counts

Pages 28 pdf

Theorems 2 source

Lemmas 12 source

Propositions 9 source

Corollaries 2 source

Definitions 4 source

Displayed equations 185 source

Bibliography entries 16 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.