Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A deformation of Borel equivariant homotopy

Gabriel Angelini-Knoll, Mark Behrens, Eva Belmont, Hana Jia Kong

TL;DR

This work develops a deformation of the $G$-equivariant stable homotopy framework by formulating complete artificial motivic categories $igl( ext{Sp}^{BG}_{ip}igr)^{ ext{Fil}};\mathbb{S}^{ ext{def}}_{G,p}$ and introducing the modified Adams--Novikov spectral sequence (MANSS). For $G=C_2$, the deformation recovers the $a$-completed Artin-- Tate Real motivic category and clarifies the relationship to the $C_2$-effective slice spectral sequence, while providing a practical computational tool via the MANSS, which blends Adams--Novikov and homotopy fixed point data in an $RO(G)$-graded, Mackey-functor-valued setting. The paper develops the foundations of filtered objects and décalage, defines tower functors, and constructs the deformation within the filtered-spectra framework, enabling collapses in key cases (e.g., $MU_ ext{R}$-projective inputs and trivial actions) and explicit $E_2$-term descriptions. Through detailed computations for $ko_{C_2}$ and $ ext{tmf}(2)$, the MANSS is shown to reproduce known slice and HFPSS phenomena, validating the approach and suggesting broad applicability to other finite groups. The results offer a new computational paradigm for RO$(G)$-graded equivariant homotopy via a motivic-like filtration, with potential implications for a range of equivariant and motivic contexts.

Abstract

We describe a deformation of the $\infty$-category of Borel $G$-spectra for a finite group $G$. This provides a new presentation of the $a$-complete real Artin--Tate motivic stable homotopy category when $G=C_2$ and gives a new interpretation of the $a$-completed $C_2$-effective slice spectral sequence. As a new computational tool, we present a modified Adams--Novikov spectral sequence which computes the $RO(G)$-graded Mackey functor valued homotopy of Borel $G$-spectra.

A deformation of Borel equivariant homotopy

TL;DR

This work develops a deformation of the -equivariant stable homotopy framework by formulating complete artificial motivic categories and introducing the modified Adams--Novikov spectral sequence (MANSS). For , the deformation recovers the -completed Artin-- Tate Real motivic category and clarifies the relationship to the -effective slice spectral sequence, while providing a practical computational tool via the MANSS, which blends Adams--Novikov and homotopy fixed point data in an -graded, Mackey-functor-valued setting. The paper develops the foundations of filtered objects and décalage, defines tower functors, and constructs the deformation within the filtered-spectra framework, enabling collapses in key cases (e.g., -projective inputs and trivial actions) and explicit -term descriptions. Through detailed computations for and , the MANSS is shown to reproduce known slice and HFPSS phenomena, validating the approach and suggesting broad applicability to other finite groups. The results offer a new computational paradigm for RO-graded equivariant homotopy via a motivic-like filtration, with potential implications for a range of equivariant and motivic contexts.

Abstract

We describe a deformation of the -category of Borel -spectra for a finite group . This provides a new presentation of the -complete real Artin--Tate motivic stable homotopy category when and gives a new interpretation of the -completed -effective slice spectral sequence. As a new computational tool, we present a modified Adams--Novikov spectral sequence which computes the -graded Mackey functor valued homotopy of Borel -spectra.
Paper Structure (27 sections, 52 theorems, 330 equations)

This paper contains 27 sections, 52 theorems, 330 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. The main constructions
  4. Computing the MANSS
  5. Computational examples
  6. Future work
  7. Organization of the paper
  8. Conventions and notation
  9. Acknowledgements
  10. Filtered objects and deformations
  11. Filtered objects
  12. Tower functors
  13. Décalage
  14. Deformations
  15. A tale of two spectral sequences
  16. ...and 12 more sections

Key Result

Theorem 1.2.1

Applying the construction (*) to the Adams--Novikov filtration with $\tau$ as the Postnikov filtration recovers the cellular $\mathbb{C}$-motivic category $\mathrm{SH}^{\text{cell}}(\mathbb{C})_{i2}$.

Theorems & Definitions (110)

  • Theorem 1.2.1: GIKR
  • Theorem 1.2.2: BHS
  • Definition : Definition \ref{['def:main']}
  • Theorem : Theorem \ref{['ESSS is MANSS']}
  • Theorem : Theorem \ref{['thm: MANSS Galois reconstruction']}
  • Theorem : Theorems \ref{['thm:MANSS=MU/a-Adams']}
  • Theorem : Theorem \ref{['thm:AHSS']}
  • Remark 2.1.3
  • Lemma 2.1.5
  • proof
  • ...and 100 more