Augmented Γ-spaces, the stable rank filtration, and a bu analogue of the Whitehead conjecture

Abstract

We explore connections between (4), where we constructed that interpolate between bu and HZ, and earlier work of Kuhn and Priddy on the Whitehead conjecture and of Rognes on the stable rank filtration in algebraic K-theory. We construct a complex of spectra that is a of an auxiliary complex used by Kuhn- Priddy; we conjecture that this chain complex is exact; and we give some supporting evidence. We tie this to work of Rognes by showing that our auxiliary complex can be constructed in terms of the stable rank filtration. As a by-product, we verify for the case of topological complex K-theory a conjecture made by Rognes about the connectiv- ity (for certain rings) of the filtration subquotients of the stable rank filtration of algebraic K-theory. In (4), we introduced a sequence of spectra{Am} interpolating between the connective complex K-theory spectrum bu and the integral Eilenberg- MacLane spectrum HZ. These new resulted from a general con- struction on permutative categories endowed with an augmentation. In the current work, we explore connections of that construction to other set- tings, in particular to work of Kuhn and Priddy (9) on the Whitehead Con- jecture, and to work of Rognes (15) on the stable rank filtration of algebraic K-theory. Connections to Kuhn and Priddy's work were suggested by the many properties that the Am share with the symmetric powers of the sphere spectrum, Sp m (S), which can also be given as an example of the categorical construction of (4). This led us to call Am the bu-analogue of Sp m (S) and to propose a of the Whitehead Conjecture. On the other hand, our construction was also reminiscent of the stable rank filtration in algebraic K-theory, and this made it natural to ask the exact relationship between the two filtrations for bu. Curiously, the two threads converged: in this paper we construct a of an auxiliary complex 2000 Mathematics Subject Classification. 2010 MSC: Primary 55P48, 19L41 ; Sec- ondary 55P91, 55R45.