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Reformulation of the stable Adams conjecture

Eric M. Friedlander

TL;DR

This work develops a rigorous, $ ext{Z}/\ell$-completed $ ext{F}$-space framework to prove a Stable Adams Conjecture in the stable homotopy category, correcting orientation issues that plagued earlier attempts. By combining rigid étale homotopy types, $ ext{Z}/\ell$-completion, and a universal representation of $X$-fibrations, it relates maps from an $ ext{F}$-space to classifying spaces $\underline{\mathcal{B}}G_\ell(X)$ with fiber-homotopy equivalence of $ ext{Z}/\ell$-completed fibrations, ultimately identifying the stable behavior of $J$-homomorphisms and Adams operations after ell-completion. The analysis hinges on the representability of $ ext{X}$-fibrations, oriented and non-oriented refinements, and comparison theorems between étale and analytic/topological models, yielding a corrected, orientation-aware proof of the Stable Adams Conjecture and a clearer link between algebraic geometry and stable homotopy theory. This advances the interface between étale homotopy theory and stable homotopy, providing concrete tools for further exploration of algebraic models of spectra and their operations.

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 étale homotopy theory. Consideration of special $\mathcal F$-spaces and together with Bousfield-Kan $\mathbb Z/\ell$-completion enables us to employ an "étale 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 $π_{X,\ell}: \underline {\mathcal B} (G_\ell(X),X_\ell) \to \underline {\mathcal B} G_\ell(X)$ determine homotopy equivalent maps of spectra if and only they correspond via pull-back of $π_{X,\ell}$ 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.

Reformulation of the stable Adams conjecture

TL;DR

This work develops a rigorous, -completed -space framework to prove a Stable Adams Conjecture in the stable homotopy category, correcting orientation issues that plagued earlier attempts. By combining rigid étale homotopy types, -completion, and a universal representation of -fibrations, it relates maps from an -space to classifying spaces with fiber-homotopy equivalence of -completed fibrations, ultimately identifying the stable behavior of -homomorphisms and Adams operations after ell-completion. The analysis hinges on the representability of -fibrations, oriented and non-oriented refinements, and comparison theorems between étale and analytic/topological models, yielding a corrected, orientation-aware proof of the Stable Adams Conjecture and a clearer link between algebraic geometry and stable homotopy theory. This advances the interface between étale homotopy theory and stable homotopy, providing concrete tools for further exploration of algebraic models of spectra and their operations.

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 étale homotopy theory. Consideration of special -spaces and together with Bousfield-Kan -completion enables us to employ an "étale functor" which commutes up to homotopy with products of simplicial schemes. In order to prove the Stable Adams Conjecture, we construct the universal -completed -fibrations for various pointed simplicial sets . Thus, two maps from a given -space to the base -space of the universal -completed -fibration determine homotopy equivalent maps of spectra if and only they correspond via pull-back of to fiber homotopy equivalent -completed -fibrations over . For the proof of the Stable Adams Conjecture, we consider maps of -spaces where is an -space model of connective -completed connective -theory.
Paper Structure (11 sections, 44 theorems, 74 equations)

This paper contains 11 sections, 44 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Frobenius maps and orientations
  3. $\mathbb Z/\ell$-completions and rigid étale homotopy types
  4. Some aspects of $\mathcal{F}$-spaces
  5. $X$-fibrations over $\mathcal{F}$-spaces
  6. $\mathbb Z/\ell$-completed $X$-fibrations
  7. Oriented $\mathbb Z/\ell$-completed $X$-fibrations
  8. Variants of the $J$-homomorphism
  9. Adams operations
  10. The Stable Adams Conjecture
  11. Concluding remark about past arguments

Key Result

Theorem 1

Let ${\bf kU}$ denote the 0-connected spectrum of (topological) complex K-theory and let ${\bf BS^{2}_\ell}$ denote a 0-connected spectrum naturally constructed using self-equivalences of $\mathbb Z/\ell$-completions of even spheres. If $p$ and $\ell$ are distinct primes, then the maps of spectra are homotopy equivalent.

Theorems & Definitions (89)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Proposition 1.1
  • Proposition 1.2
  • Corollary 1.3
  • proof
  • Definition 1.4
  • Proposition 1.5
  • proof
  • ...and 79 more