Augmented Gamma-spaces, the stable rank filtration, and a bu-analogue of the Whitehead Conjecture

TL;DR

This paper constructs a chain complex of spectra as a bu-analogue of Kuhn-Priddy's auxiliary complex, explores its conjectured exactness, and connects it to Rognes' work on the stable rank filtration, providing evidence and verifying related conjectures.

Contribution

It introduces a new chain complex of spectra linking bu and HZ, conjectures its exactness, and relates it to the stable rank filtration in algebraic K-theory.

Findings

Constructed a bu-analogue of Kuhn-Priddy's auxiliary complex.

Provided evidence supporting the conjecture of exactness.

Verified Rognes' conjecture on the connectivity of filtration subquotients.

Abstract

We explore connections between our earlier work, in which we constructed spectra 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 "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.