A multiplicative version of the tom Dieck splitting

2025/05/05 by Andrew J. Blumberg, Michael A. Mandell, Blumberg, Andrew J. +1
Medicine · #55P91 #Algebraic Topology (math.AT) #FOS: Mathematics #Medical Imaging Techniques and Applications #Primary 55P43

paper · pdf · doi:10.48550/arxiv.2505.02721

openalex publication_date 2025/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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_∞ ring spectra. Just as the usual tom Dieck splitting characterizes the equivariant stable category associated to an N_∞ operad N, the multiplicative tom Dieck splitting characterizes the G-symmetric monoidal structure on the genuine equivariant stable category associated to N.

Citations

• The homotopy theory of cyclotomic spectra, 10 years later
• Free duals and a new universal property for stable equivariant homotopy theory
• Norms for compact Lie groups in equivariant stable homotopy theory
• Equivariant infinite loop space theory, the space level story
• Operadic multiplications in equivariant spectra, norms, and transfers
• On the non-existence of elements of Kervaire invariant one
• On the nonexistence of elements of Kervaire invariant one

Related

• The tom Dieck splitting theorem in equivariant motivic homotopy theory
• The homotopy theory of cyclotomic spectra, 10 years later
• Stratified categories, geometric fixed points and a generalized Arone-Ching theorem
• Spectra and symmetric spectra in general model categories
• Multiplicative Properties of the Slice Filtration