A multiplicative version of the tom Dieck splitting

TL;DR

This paper extends the classical tom Dieck splitting to a multiplicative setting for derived pushforwards of $N_{ abla}$ ring spectra, revealing a new structure in equivariant stable homotopy theory.

Contribution

It introduces a multiplicative version of the tom Dieck splitting formula for $N_{ abla}$ ring spectra, linking it to the $G$-symmetric monoidal structure.

Findings

Established a multiplicative splitting formula for derived pushforwards.

Connected the splitting to the $G$-symmetric monoidal structure.

Generalized classical splitting to a multiplicative context.

Abstract

While the classical tom Dieck splitting in equivariant stable homotopy theory is typically regarded as a formula for suspension spectra in the genuine equivariant stable category, it can be interpreted as a calculation of the fixed points of $G$-spectra that are derived pushforwards from the naive equivariant stable category. We then establish a corresponding multiplicative splitting formula for derived pushforwards of $N_{\infty}$ ring spectra. Just as the usual tom Dieck splitting characterizes the equivariant stable category associated to an $N_{\infty}$ operad $\mathcal{N}$, the multiplicative tom Dieck splitting characterizes the $G$-symmetric monoidal structure on the genuine equivariant stable category associated to $\mathcal{N}$.