Dyer-Lashof-Araki-Kudo operations

These are operations on the mod-$p$ homology of infinite loop spaces. They measure the failure of the Pontrjagin product to be commutative at the chain-level and are analogous to the Steenrod squares. The operations are linear maps of the form

$$\beta^\epsilon Q^r: H_n(X;\mathbb{F}_p) \to H_{n + 2r(p-1) -\epsilon}(X;\mathbb{F}_p)$$

for $\epsilon \in \{0,1\}$ and $r \in \mathbb{Z}_{\geq \epsilon}$.