The subgroup stratification of Nakaoka spectra

TL;DR

This paper introduces a stratification of Nakaoka spectra for G-Tambara functors, revealing new properties of tale maps and their topological behavior in the equivariant setting, especially for Dedekind groups and specific examples.

Contribution

It constructs a stratification of Nakaoka spectra for G-Tambara functors and analyzes the topological properties of tale maps in the equivariant context, providing new insights.

Findings

The Hth stratum of the Nakaoka spectrum of the Burnside G-Tambara functor is closed and not open.

Examples of tale maps induce closed, non-open maps on Nakaoka spectra.

Computed strata of fixed-point Tambara functors and ghost of C_p-Tambara functors.

Abstract

We construct a stratification on the Nakaoka spectrum of any $G$-Tambara functor indexed by the poset of subgroups of $G$. When $G$ is Dedekind, we show that the $H$th stratum of the Nakaoka spectrum of the Burnside $G$-Tambara functor is closed and not open; this provides examples of \'{e}tale maps which induce closed, non-open maps on Nakaoka spectra. By computing the strata on the ghost of a $C_p$-Tambara functor we obtain many examples of \'{e}tale maps of Tambara functors for which the induced map on Nakaoka spectra is not open, in contrast to the non-equivariant world. We also compute the strata of all fixed-point Tambara functors.