  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 $R$ is between $\bZ$ and $\bQ$, with the $x$-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 $R \llbrack x \rrbrack$
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
$\bQ$-algebras.
