Moduli space of filtered lambda-ring structures over a filtered ring

Motivated by recent works on the genus of classifying spaces of compact Lie groups, here we study the set of filtered $\lambda$-ring structures over a filtered ring from a purely algebraic point of view. From a global perspective, we first show that this set has a canonical topology compatible with the filtration on the given filtered ring. For power series rings $R \llbrack x \rrbrack$,where$R$isbetween$\bZ$and$\bQ$,withthe$x$-adic filtration, we mimic the construction of the Lazard ring in formal group theory and show that the set of filtered $\lambda$-ring structures over $R\text{llbrack}x\text{rrbrack}$ is canonically isomorphic to the set of ring maps from some ``universal’’ ring $U$ to $R$. From a local perspective, we demonstrate the existence of uncountably many mutually non-isomorphic filtered $\lambda$-ring structures over some filtered rings, including rings of dual numbers over binomial domains, (truncated) polynomial and powers series rings over torsionfree $\text{bQ}$-algebras.