Augmented $\mit\Gamma$-spaces, the stable rank filtration, and a $bu$ analogue of the Whitehead conjecture
We explore connections between our previous paper [J. Reine Angew. Math. 604 (2007)], where we constructed spectra that interpolate between $bu$ and $\rm H\mathbb Z$, 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 “chain complex of spectra” that is a $bu$ analogue 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 connectivity (for certain rings) of the filtration subquotients of the stable rank filtration of algebraic $K$-theory.