A multiplicative version of the tom Dieck splitting
Andrew J. Blumberg, Michael A. Mandell
TL;DR
This work provides a multiplicative analogue of the tom Dieck splitting in genuine $G$-equivariant stable homotopy theory by analyzing derived pushforwards of $N_{\infty}$-algebras. It formulates a natural split for the derived geometric fixed points of a pushforward $R$, namely $\displaystyle \coprod_{(K)<G} R^{\Phi K}\otimes^{\mathrm{L}}_{WK}EWK \simeq (j^{\mathrm{L}}_{*}R)^{\Phi G}$ in the homotopy category of $E_{\infty}$-algebras in non-equivariant spectra, with the coproduct indexed by admissible subgroups determined by the indexing system. The construction relies on $N_{\infty}$ operad theory, indexing systems, and detailed computations of geometric fixed points for free $\mathcal{N}$-algebras, culminating in a proof that connects additive and multiplicative phenomena via derived fixed-point data. These results distinguish between $G$-symmetric monoidal structures corresponding to different $N_{\infty}$-operads and generalize to compact Lie groups through an operadic and fixed-point framework, providing a robust multiplicative counterpart to the classical tom Dieck splitting.
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 $\mathcal{N}$.