Reformulation of the stable Adams conjecture

Abstract

We revisit methods of proof of the Adams Conjecture in order to correct and supplement earlier efforts to prove analogous conjectures in the stable homotopy category. We utilize simplicial schemes over an algebraically closed field of positive characteristic and a rigid version of Artin-Mazur \'etale homotopy theory. Consideration of special $\mathcal F$-spaces and together with Bousfield-Kan $\mathbb Z/\ell$-completion enables us to employ an "\'etale functor" which commutes up to homotopy with products of simplicial schemes. In order to prove the Stable Adams Conjecture, we construct the universal $\mathbb Z/\ell$-completed $X$-fibrations for various pointed simplicial sets $X$. Thus, two maps from a given $\mathcal F$-space $\underline{\mathcal B}$ to the base $\mathcal F$-space of the universal $\mathbb Z/\ell$-completed $X$-fibration $\pi_{X,\ell}: \underline {\mathcal B} (G_\ell(X),X_\ell) \to \underline {\mathcal B} G_\ell(X)$ determine homotopy equivalent maps of spectra if and only they correspond via pull-back of $\pi_{X,\ell}$ to fiber homotopy equivalent $\mathbb Z/\ell$-completed $X$-fibrations over $\underline {\mathcal B}$. For the proof of the Stable Adams Conjecture, we consider maps of $\mathcal F$-spaces $\underline {\mathcal B }\to \underline {\mathcal B} G_\ell(S^2)$ where $\underline {\mathcal B}$ is an $\mathcal F$-space model of connective $\ell$-completed connective $K$-theory.