Modular coinvariants and the mod p homology of QS^k

TL;DR

This paper uses modular invariant theory to fully describe the mod p homology of the infinite loop spaces QS^k as a coalgebraic ring, including the actions of Dyer-Lashof and Steenrod algebras.

Contribution

It establishes a complete set of relations for the mod p homology of QS^k as a coalgebraic ring for odd primes, advancing understanding of its algebraic structure.

Findings

Complete relations for mod p homology of QS^k as a coalgebraic ring

Explicit description of Dyer-Lashof and Steenrod algebra actions

Enhanced understanding of algebraic structures in stable homotopy theory

Abstract

We use modular invariant theory to establish a complete set of relations of the mod $p$ homology of $\{QS^k\}_{k\geq0}$, for $p$ odd, as a ring object in the category of coalgebras (also known as a coalgebraic ring or a Hopf ring). We also describe the action of the mod $p$ Dyer-Lashof algebra as well as the mod $p$ Steenrod algebra on the coalgebraic ring.