Symmetric products and subgroup lattices

TL;DR

This paper establishes a connection between the rational homotopy groups of symmetric products of the G-equivariant sphere spectrum and the rational homology of specific subcomplexes of G's subgroup lattice, revealing new algebraic-topological relationships.

Contribution

It introduces a novel isomorphism linking equivariant stable homotopy groups with subgroup lattice homology, advancing understanding of equivariant spectra.

Findings

Rational homotopy groups correspond to subgroup lattice homology.

Identifies specific subcomplexes of the subgroup lattice relevant to symmetric products.

Provides a new algebraic-topological framework for equivariant spectra.

Abstract

Let G be a finite group. We show that the rational homotopy groups of symmetric products of the G-equivariant sphere spectrum are naturally isomorphic to the rational homology groups of certain subcomplexes of the subgroup lattice of G.