Slice spectral sequences through synthetic spectra

TL;DR

This paper introduces a new t-structure on filtered G-spectra, linking slice and homotopy fixed-point filtrations, and explores the refined algebraic structures in the slice spectral sequence for Real bordism theory, with implications for equivariant stable homotopy theory.

Contribution

It defines a t-structure on filtered G-spectra that unifies slice and homotopy fixed-point filtrations, and shows how the slice spectral sequence refines to an E-infinity algebra in MU-synthetic spectra.

Findings

Slice spectral sequence refines to an E-infinity algebra in MU-synthetic spectra.

A map of multiplicative spectral sequences respects higher E-infinity structures.

Conditional conjecture on vanishing lines suggests further algebraic lifting in equivariant spectra.

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 $G$.