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Uniqueness of real ring spectra up to higher homotopy

Jack Morgan Davies

TL;DR

The paper develops a framework for uniqueness of maps between low-height $ ext{E}_ ext∞-rings using a real-variant of Goerss–Hopkins obstruction theory. It defines real $ heta$-algebras and real spectra, and shows that many elliptic cohomology theories (e.g., $ ext{tmf}$, $ ext{TMF}_0(N)$, $ ext{KO}[[q]]$) are real, enabling obstruction-theoretic control of maps to $K$-theory-like theories. It proves that the $p$-adic $q$-expansion map from elliptic cohomology to topological $K$-theory is uniquely determined by zeroth $p$-adic KO- or $K$-homology up to $3$-homotopy, with stronger fixed-point statements when passing to $p$-adic Galois fixed points, and derives a coherent higher functoriality of elliptic Adams operations, including the construction of maps $oldsymbol{ abla} o ext{CAlg}$ that realize $oldsymbol{ abla}^k$-actions up to controlled homotopies. The work also indicates applications to connective models of Behrens' $Q(N)$ spectra and broadens the toolbox for chromatic homotopy theory, connecting modular and $K$-theoretic data via real obstruction theory.

Abstract

We discuss a notion of uniqueness up to $n$-homotopy and study examples from stable homotopy theory. In particular, we show that the $q$-expansion map from elliptic cohomology to topological $K$-theory is unique up to $3$-homotopy, away from the prime $2$, and that upon taking $p$-completions and $\mathbf{F}_p^\times$-homotopy fixed points, this map is uniquely defined up to $(2p-3)$-homotopy. Using this, we prove new relationships between Adams operations on connective and dualisable topological modular forms -- other applications, including a construction of a connective model of Behrens' $Q(N)$ spectra away from $2N$, will be explored elsewhere. The technical tool facilitating this uniqueness is a variant of the Goerss--Hopkins obstruction theory for real spectra, which applies to various elliptic cohomology and topological $K$-theories with a trivial complex conjugation action as well as some of their homotopy fixed points.

Uniqueness of real ring spectra up to higher homotopy

TL;DR

The paper develops a framework for uniqueness of maps between low-height heta ext{tmf} ext{TMF}_0(N) ext{KO}[[q]]KpqKpK3poldsymbol{ abla} o ext{CAlg}oldsymbol{ abla}^kQ(N)K$-theoretic data via real obstruction theory.

Abstract

We discuss a notion of uniqueness up to -homotopy and study examples from stable homotopy theory. In particular, we show that the -expansion map from elliptic cohomology to topological -theory is unique up to -homotopy, away from the prime , and that upon taking -completions and -homotopy fixed points, this map is uniquely defined up to -homotopy. Using this, we prove new relationships between Adams operations on connective and dualisable topological modular forms -- other applications, including a construction of a connective model of Behrens' spectra away from , will be explored elsewhere. The technical tool facilitating this uniqueness is a variant of the Goerss--Hopkins obstruction theory for real spectra, which applies to various elliptic cohomology and topological -theories with a trivial complex conjugation action as well as some of their homotopy fixed points.
Paper Structure (10 sections, 17 theorems, 48 equations)

This paper contains 10 sections, 17 theorems, 48 equations.

Table of Contents

  1. Definitions and notation
  2. Uniqueness up to $n$-homotopy
  3. $\mathop{\mathrm{K}}\nolimits(1)$-local homology theories and comodules
  4. Real spectra and their Goerss--Hopkins obstruction theory
  5. Real $\psi$-modules and $\bar{\psi}$-modules
  6. Real spectra and their Goerss--Hopkins spectral sequence (\ref{['ghtheorymain']})
  7. Uniqueness of maps of between real $\mathbf{E}_\infty$-rings
  8. Real elliptic cohomology and $K$-theories
  9. Uniqueness of $p$-adic topological $q$-expansion map (\ref{['mainuniquenessstatement', 'mainuniquenessstatementpgrowth']})
  10. Higher functoriality of elliptic Adams operations (\ref{['adamsoperationshomotopiesintegral', 'adamsoperationshomotopiespadic']})

Key Result

Theorem A

A map of $\mathbf{E}_\infty$-rings $\mathop{\mathrm{tmf}}\nolimits \to \mathop{\mathrm{KO}}\nolimits\llbracket q\rrbracket[\frac{1}{2}]$ is uniquely determined by its effect upon applying complex $K$-theory homology in degree $0$, up to $3$-homotopy.As one will see in our proof of mainuniquenessstat

Theorems & Definitions (45)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Definition 1.1
  • Proposition 1.2
  • proof
  • Definition 1.3
  • Definition 1.4
  • ...and 35 more