String bordism and chromatic characteristics. (English) Zbl 1427.55004
Davis, Daniel G. (ed.) et al., Homotopy theory: tools and applications. A conference in honor of Paul Goerss’s 60th birthday, University of Illinois at Urbana-Champaign, Urbana, IL, USA, July 17–21, 2017. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 729, 239-254 (2019).
The author gives a natural extension of his notion of characteristics for ring spectra [M. Szymik, Algebr. Geom. Topol. 14, No. 6, 3717–3743 (2014; Zbl 1311.55014)] to chromatic homotopy theory, using the Hopkins-Miller classes \(\zeta_n\in \pi_{-1} (\hat{\mathbb{S}}) \) in the homotopy of the \(K(n)\)-local sphere, where \(K(n)\) is the \(n\)th Morava \(K\)-theory (at an implicit prime \(p\)). A \(K(n)\)-local ring spectrum \(A\) is said to have chromatic characteristic \(\zeta_n\) if \((\eta_A)_* \zeta_n = 0 \) in \(\pi_{-1} (A)\), where \(\eta_A\) is the unit of \(A\).
The author’s principal interest is in the \(E_\infty\) ring spectra case, motivated by examples related to bordism. To this end, he constructs \(\hat{\mathbb{S}} /\!\!/ \zeta_n\), the versal \(K(n)\)-local \(E_\infty\) ring spectrum of characteristic \(\zeta_n\).
The notion of chromatic characteristics is illustrated by further examples. The \(K(1)\)-localizations of the connective \(K\)-theory spectra \(ko\) and \(ku\) are shown to have characteristic \(\zeta_1\), whereas, in coprime characteristic, the \(K(1)\)-localization of the algebraic \(K\)-theory spectrum \(K(\mathbb{F}_q)\) does not.
Likewise, the \(K(2)\)-localization of the topological modular forms spectrum \(tmf\) is shown to have characteristic \(\zeta_2\) and, for all \(n\geq 1\), the \(K(n)\)-localization of the spin cobordism spectrum \(\mathrm{MSpin}\) to have characteristic \(\zeta_n\).
For the entire collection see [Zbl 1419.55001].
The author’s principal interest is in the \(E_\infty\) ring spectra case, motivated by examples related to bordism. To this end, he constructs \(\hat{\mathbb{S}} /\!\!/ \zeta_n\), the versal \(K(n)\)-local \(E_\infty\) ring spectrum of characteristic \(\zeta_n\).
The notion of chromatic characteristics is illustrated by further examples. The \(K(1)\)-localizations of the connective \(K\)-theory spectra \(ko\) and \(ku\) are shown to have characteristic \(\zeta_1\), whereas, in coprime characteristic, the \(K(1)\)-localization of the algebraic \(K\)-theory spectrum \(K(\mathbb{F}_q)\) does not.
Likewise, the \(K(2)\)-localization of the topological modular forms spectrum \(tmf\) is shown to have characteristic \(\zeta_2\) and, for all \(n\geq 1\), the \(K(n)\)-localization of the spin cobordism spectrum \(\mathrm{MSpin}\) to have characteristic \(\zeta_n\).
For the entire collection see [Zbl 1419.55001].
Reviewer: Geoffrey Powell (Angers)
MSC:
| 55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |
| 55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |
| 19L41 | Connective \(K\)-theory, cobordism |
| 57R90 | Other types of cobordism |
| 58J26 | Elliptic genera |
Citations:
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