On adic genus and lambda-rings

Abstract

Abstract. Sufficient conditions on a space are given which guarantee that the K-theory ring is an invariant of the adic genus. An immediate consequence of this result about adic genus is that for any positive integer n, the power series ring Z[[x 1 , . . . , xn]] admits uncountably many pairwise non-isomorphic λ-ring structures.

Moduli space of filtered λ‐ringstructures over a filtered ring

“…It should also be remarked that this ring U is non-trivial, since the power series filtered ring Z[[x]] admits uncountably many mutually non-isomorphic filtered λ-ring structures (see [14]).…”

“…Note that this map p is injective if and only if the intersection ∩ n≥1 I n is 0. This has relevance, for example, in topology: If X is a space in the genus of a torsionfree classifying space BG of a simply-connected compact Lie group G, then its integral K-theory K(X) is complete Hausdorff (see [14]).…”

“…In each case we will exhibit uncountably many mutually non-isomorphic filtered λ-ring structures, which in particular implies that the moduli space has uncountably many points. Unlike in [14] where the existence of uncountably many mutually non-isomorphic λ-ring structures over the power series ring Z[[x 1 , . .…”

“…A case in point is the genus of the classifying space BSU (2). As the author shows in [14], spaces in its genus (which there are uncountably many of them up to homotopy, see [10]) all have the same K-theory filtered ring, Z[[x]] with x in filtration precisely 4, up to isomorphism. Therefore, for each one of these K-theory filtered λ-rings, the Adams operation ψ p , p any prime, satisfies the congruence condition ψ p (x) ≡ p 2 x (mod x 2 ).…”

“…The author showed in [14] that K-theory filtered ring is an invariant of the genus of a space X, provided X is simply-connected and has even and torsionfree integral homology and a finitely generated power series K-theory filtered ring. Here the filtration in K-theory comes from a skeletal filtration of the space.…”