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Slice spectral sequences through synthetic spectra

Christian Carrick

TL;DR

This work constructs a t-structure on filtered G-spectra in which the equivariant slice filtration is the connective cover of the homotopy fixed-point filtration, unifying two central filtrations in equivariant stable homotopy theory. It then leverages this framework to show that the slice spectral sequence for Real bordism norms MU^{(G)} refines to an E_1∞-algebra in MU-synthetic spectra for cyclic 2-groups, and produces a multiplicative map from the classical Adams–Novikov spectral sequence to the slice spectral sequence, preserving higher E_1∞-structure. The paper develops RO(G)-indexed filtrations and Whitehead-type towers, and introduces linear t-structures that govern connective behavior, enabling both integer- and RO(G)-graded analyses. Conditional on vanishing-line conjectures in the MU_G-based Adams spectral sequence, these constructions lift the slice filtration to an O-algebra in MU_G-synthetic spectra, providing a robust framework for structured, normed equivariant chromatic computations and potential connections to motivic and elliptic-curve theories. Together, these results offer a flexible, multiplicative approach to understanding equivariant slices, with concrete computational leverage via synthetic categories and RO(G)-graded filtrations, and set a program for further unifications across motivic and chromatic contexts.

Abstract

We define a $t$-structure on the category of filtered $G$-spectra such that for a Borel $G$-spectrum $X$ the slice filtration of $X$ is the connective cover of the homotopy fixed-point filtration of $X$. Using this, we show that the slice spectral sequence for the norm $N_{C_2}^GMU_{\mathbb{R}}$ of Real bordism theory refines canonically to a $\mathbb{E}_\infty$-algebra in $MU$-synthetic spectra, when $G$ is a cyclic $2$-group. Concretely, this gives a map of multiplicative spectral sequences from the classical Adams--Novikov spectral sequence of $\mathbb{S}$ to the slice spectral sequence for $N_{C_2}^GMU_{\mathbb{R}}$ that respects the higher $\mathbb{E}_\infty$ structure, such as Toda brackets and power operations. We give a conjecture on the existence of vanishing lines in the equivariant Adams--Novikov spectral sequence based at tom Dieck's homotopical complex bordism $MU_G$. Conditional on this conjecture, our $t$-structure implies that the slice filtration for $N_{C_2}^GMU_{\mathbb{R}}$ lifts further to an $\mathcal{O}$-algebra in $MU_G$-synthetic spectra, where $\mathcal{O}$ is the $\mathbb{N}_\infty$-operad with all norms from nontrivial subgroups of $G$.

Slice spectral sequences through synthetic spectra

TL;DR

This work constructs a t-structure on filtered G-spectra in which the equivariant slice filtration is the connective cover of the homotopy fixed-point filtration, unifying two central filtrations in equivariant stable homotopy theory. It then leverages this framework to show that the slice spectral sequence for Real bordism norms MU^{(G)} refines to an E_1∞-algebra in MU-synthetic spectra for cyclic 2-groups, and produces a multiplicative map from the classical Adams–Novikov spectral sequence to the slice spectral sequence, preserving higher E_1∞-structure. The paper develops RO(G)-indexed filtrations and Whitehead-type towers, and introduces linear t-structures that govern connective behavior, enabling both integer- and RO(G)-graded analyses. Conditional on vanishing-line conjectures in the MU_G-based Adams spectral sequence, these constructions lift the slice filtration to an O-algebra in MU_G-synthetic spectra, providing a robust framework for structured, normed equivariant chromatic computations and potential connections to motivic and elliptic-curve theories. Together, these results offer a flexible, multiplicative approach to understanding equivariant slices, with concrete computational leverage via synthetic categories and RO(G)-graded filtrations, and set a program for further unifications across motivic and chromatic contexts.

Abstract

We define a -structure on the category of filtered -spectra such that for a Borel -spectrum the slice filtration of is the connective cover of the homotopy fixed-point filtration of . Using this, we show that the slice spectral sequence for the norm of Real bordism theory refines canonically to a -algebra in -synthetic spectra, when is a cyclic -group. Concretely, this gives a map of multiplicative spectral sequences from the classical Adams--Novikov spectral sequence of to the slice spectral sequence for that respects the higher structure, such as Toda brackets and power operations. We give a conjecture on the existence of vanishing lines in the equivariant Adams--Novikov spectral sequence based at tom Dieck's homotopical complex bordism . Conditional on this conjecture, our -structure implies that the slice filtration for lifts further to an -algebra in -synthetic spectra, where is the -operad with all norms from nontrivial subgroups of .
Paper Structure (23 sections, 46 theorems, 193 equations, 2 figures)

This paper contains 23 sections, 46 theorems, 193 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Notation and conventions
  4. Outline of results and methods
  5. The slice spectral sequence and connective models
  6. Linear $t$-structures and slice filtrations
  7. Maps from the $MU_G$-based ANSS
  8. $RO(G)$-indexed filtrations
  9. Questions
  10. Equivariant filtered objects
  11. $G$-symmetric monoidal $\infty$-categories and geometric fixed points
  12. Equivariant filtered objects
  13. Geometric fixed points and dilation
  14. The $RO(G)$-graded Whitehead filtration and equivariant $t$-structures
  15. Filtered $t$-structures and slice filtrations
  16. ...and 8 more sections

Key Result

Theorem 1.1

There is an accessible $t$-structure $\tau_{\ge0}^{\mathrm{slice}}$ on $\mathrm{Fil}(\mathcal{S}p^G)$ that is compatible with the $G$-symmetric monoidal structure such that, for any $X\in \mathcal{S}p^G$,

Figures (2)

  • Figure 1: The ANSS for $ko$.
  • Figure 2: The $\tau$-BSS for $\tau_{\ge0}^{y=(1/2)x}\mathrm{AN}(ko)$.

Theorems & Definitions (119)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • proof : proof of Theorem \ref{['thm:introthmsyntehtic']}
  • Remark 2.2
  • ...and 109 more