On λ‐rings and topological realization

Abstract

It is shown that most possibly truncated power series rings admit uncountably many filtered λ-ring structures. The question of how many of these filtered λ-ring structures are topologically realizable by the K-theory of torsion-free spaces is also considered for truncated polynomial rings.

Cohomology of λ-rings

“…To finish the proof, we only need to observe that if (a, b) ∈ H 0 λ (R) and In this section, we will compute the algebra H 0 λ (R) for each of the uncountably many isomorphism classes of filtered λ-ring structures on Z[x]/(x 3 ), with x in some fixed positive filtration. (See [10] for a proof of this uncountability statement.) 5.1.…”

“…Let us begin by recalling the classification of filtered λ-ring structures on Z[x]/(x 3 ). See [10] for more details.…”

“…Before we proceed, we should explain the significance of these 64 filtered λ-rings. The following statement is shown in [10]: If X is a torsionfree topological space (i.e. its integral cohomology is Z-torsionfree) whose unitary K-theory K(X), as a filtered ring, is the ring Z[x]/(x 3 ), then K(X) is isomorphic as a filtered λ-ring to one of the 64 S((p r ), h) in (6.0.5).…”

“…Among the uncountably many isomorphism classes of filtered λ-ring structures on the truncated polynomial ring Z[x]/(x 3 ), at most 64 of them can be realized as the K-theory of torsionfree spaces (see [10]). The main result of section 6 is Theorem 6.1, which describes the groups H 1 λ and determines (non-)commutativity of the graded algebras H ≤1 λ for these 64 isomorphism classes of filtered λ-rings.…”

“…Interestingly enough, exactly 35 of those 64 graded algebras H ≤1 λ (R) are commutative. It is also shown in [10] that the truncated polynomial ring Z[x]/(x 4 ) admits uncountably many isomorphism classes of filtered λ-ring structures and that, among these classes, at most 61 of them can be realized by the Ktheory of torsionfree spaces. The main results of section 7, Theorem 7.1 and Theorem 7.2, compute the algebras H 0 λ for these 61 isomorphism classes of filtered λ-rings.…”