Power operations and coactions in highly commutative homology theories

TL;DR

This paper develops new formulae for power operations and homology coactions in highly structured ring spectra, simplifying calculations in algebraic topology, especially for mod p homology and the Dyer-Lashof action.

Contribution

It introduces a unified approach to derive formulae for power operations and coactions in a broad class of spectra, extending previous results to odd primes.

Findings

Derived formulae for power operations and homology coactions.

Extended Dyer-Lashof action formulae to odd primes.

Provided sample calculations demonstrating the applicability.

Abstract

Power operations in the homology of infinite loop spaces, and $H_\infty$ or $E_\infty$ ring spectra have a long history in Algebraic Topology. In the case of ordinary mod p homology for a prime p, the power operations of Kudo, Araki, Dyer and Lashof interact with Steenrod operations via the Nishida relations, but for many purposes this leads to complicated calculations once iterated applications of these functions are required.On the other hand, the homology coaction turns out to provide tractable formulae better suited to exploiting multiplicative structure. We show how to derive suitable formulae for the interaction between power operations and homology coactions in a wide class of examples; our approach makes crucial use of modern frameworks for spectra with well behaved smash products. In the case of mod $p$ homology, our formulae extend those of Bisson and Joyal to odd primes. We also show how to exploit our results in sample calculations, and produce some apparently new formulae for the Dyer-Lashof action on the dual Steenrod algebra.