Uniqueness of real ring spectra up to higher homotopy
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916644700291072 |
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| author | Davies, Jack Morgan |
| author_facet | Davies, Jack Morgan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_02173 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniqueness of real ring spectra up to higher homotopy Davies, Jack Morgan Algebraic Topology K-Theory and Homology 55N34, 55N22, 55S25, 55P43, 14A20 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. |
| title | Uniqueness of real ring spectra up to higher homotopy |
| topic | Algebraic Topology K-Theory and Homology 55N34, 55N22, 55S25, 55P43, 14A20 |
| url | https://arxiv.org/abs/2305.02173 |