On λ-rings and topological realization

A λ-ring is, roughly speaking, a commutative ring R with unit together with operations λi, i ≥ 0, on it that act like the exterior power operations. It is widely used in algebraic topology, algebra, and representation theory. For example, the complex representation ring R(G) of a group G is a λ-ring, where λi is induced by the map that sends a representation to its ith exterior power. Another example of a λ-ring is the complex K-theory of a topological space X . Here, λi arises from the map that sends a complex vector bundle η over X to the ith exterior power of η. In the algebra side, the universal Witt ringW(R) of a commutative ring R is a λ-ring. The purpose of this paper is to consider the following two interrelated questions: (i) classify the λ-ring structures over power series and truncated polynomial rings; (ii) which ones and how many of these λ-rings are realizable as (i.e., isomorphic to) the K-theory of a topological space? The first question is purely algebraic, with no topology involved. One can think of the second question as a K-theoretic analogue of the classical Steenrod question, which asks for a classification of polynomial rings (over the field of p elements and has an action by the mod p Steenrod algebra) that can be realized as the singular mod p cohomology of a topological space. In addition to being a λ-ring, the K-theory of a space is filtered, making K(X) a filtered λ-ring. Precisely, by a filtered λ-ring wemean a filtered ring (R,{R= I0 ⊃ I1 ⊃ ···}) in which R is a λ-ring and the filtration ideals In are all closed under the λ-operations λi (i > 0). It is, therefore, more natural for us to consider filtered λ-ring structures over