Homology operations revisited

The mod p homology of E-infinity spaces is a classical topic in algebraic topology traditionally approached in terms of Dyer–Lashof operations. In this paper, we offer a new perspective on the subject by providing a detailed investigation of an alternative family of homology operations equivalent to, but distinct from, the Dyer–Lashof operations. Among other things, we will relate these operations to the Dyer–Lashof operations, describe the algebra generated by them, and use them to describe the homology of free E-infinity spaces. We will also investigate the relationship between the operations arising from the additive and multiplicative E-infinity structures on an E-infinity ring space. The operations have especially good properties in this context, allowing for a simple and conceptual formulation of “mixed Adem relations” describing how the operations arising from the two different E-infinity structures interact.