Augmented $\mit\Gamma$-spaces, the stable rank filtration, and a $bu$ analogue of the Whitehead conjecture

Gregory Z. Arone¹ ; Kathryn Lesh²

¹ Kerchof Hall University of Virginia P.O. Box 400137 Charlottesville, VA 22904, U.S.A.
² Department of Mathematics Union College Schenectady, NY 12309, U.S.A.

Fundamenta Mathematicae, Tome 207 (2010) no. 1, pp. 29-70.

Résumé

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.

Détail

DOI : 10.4064/fm207-1-3

Keywords: explore connections between previous paper reine angew math where constructed spectra interpolate between mathbb earlier work kuhn priddy whitehead conjecture rognes stable rank filtration algebraic k theory construct chain complex spectra analogue auxiliary complex kuhn priddy conjecture chain complex exact supporting evidence tie work rognes showing auxiliary complex constructed terms stable rank filtration by product verify topological complex k theory conjecture made rognes about connectivity certain rings filtration subquotients stable rank filtration algebraic k theory