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Uniqueness of real ring spectra up to higher homotopy

Jack Morgan Davies

Vol. 9 (2024), No. 3, 447–473
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 Fp×-homotopy fixed points, this map is uniquely defined up to (2p3)-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 K(1)-local Goerss–Hopkins obstruction theory for real spectra, which applies to various elliptic cohomology theories and topological K-theories with a trivial complex conjugation action as well as some of their homotopy fixed points.

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Keywords
topological $K$-theory, chromatic homotopy theory, topological modular forms, uniqueness up to homotopy, structure ring spectra, elliptic cohomology theory, topological $q$-expansion map, Adams operations, Goerss–Hopkins obstruction theory
Mathematical Subject Classification
Primary: 55N15, 55N22, 55N34, 55P43, 55S25
Secondary: 14A20
Milestones
Received: 13 June 2023
Revised: 23 June 2024
Accepted: 11 July 2024
Published: 28 August 2024
Authors
Jack Morgan Davies
Mathematisches Institut
Universität Bonn
Bonn
Germany