Synthetic equivariant spectra for finite abelian groups and motivic homotopy theory

TL;DR

This paper establishes a topological reconstruction for cellular A-equivariant motivic spectra over complex numbers, linking it to synthetic spectra and equivariant algebraic cobordism, advancing understanding in equivariant homotopy theory.

Contribution

It proves a reconstruction theorem connecting equivariant motivic spectra with synthetic spectra for finite abelian groups, using equivariant formal group laws.

Findings

Equivariant algebraic cobordism homotopy groups described via formal group laws

Reconstruction equivalence after prime completion

Synthetic spectra as deformations of equivariant spectra

Abstract

We prove a topological reconstruction result for the category of cellular $A$-equivariant motivic spectra over the complex numbers where $A$ is a finite abelian group: after completion at an arbitrary prime, this is equivalent to the completion of a category of synthetic $A$-equivariant spectra. The latter is a deformation of equivariant spectra which categorifies the equivariant perfect even filtration and is closely related to the equivariant Adams--Novikov spectral sequence. Our main computational input is a description of the bigraded homotopy groups of equivariant algebraic cobordism in terms of equivariant formal group laws.