Characteristic classes in \(TMF\) of level \(\Gamma_{1}(3)\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1345.55001
author: Gerd Laures
published in: Transactions of the American Mathematical Society
publication date: 2016-03-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.3588
Mathematics Subject Classification ID: 55N34
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 22E67
zbMATH DE Number: 6559892
zbMATH Keywords: topological modular forms
zbMATH Keywords: spin group
zbMATH Keywords: string group
zbMATH Keywords: Johnson-Wilson theories
zbMATH Keywords: Pontryagin classes
zbMATH Keywords: positive energy representation
MaRDI profile type: Publication
OpenAlex ID: W2297555861
arXiv ID: 1304.3588
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DOI: 10.1090/TRAN/6575
title: Characteristic classes in \(TMF\) of level \(\Gamma_{1}(3)\)
review text: It is a classical theorem by Wall that oriented bordism is determined by Stiefel-Whitney numbers and Pontryagin numbers. Likewise, spin bordism is determined by Stiefel-Whitney numbers and \(KO\)-Pontryagin numbers. There is hope that one might get a similar understanding of string bordism using characteristic classes based on topological modular forms (with level structures).NEWLINENEWLINEThe article under review considers characteristic classes for the spectrum \(TMF_1(3)\). This is a Landweber exact version of the spectrum of topological modular forms \(TMF\) and indeed a generalized form of a Johnson-Wilson spectrum \(E(2)\) at the prime \(2\). The author constructs Pontryagin classes for spin bundles for \(TMF_1(3)\) and shows that \(TMF_1(3)^*(BSpin)\) is freely generated by these as a power series ring over \(TMF_1(3)^*\). The method is the following: First, the author uses the calculation of \(K(2)^*(BSpin)\) from [\textit{N. Kitchloo} and \textit{G. Laures}, K-Theory 25, No. 3, 201--214 (2002; Zbl 0997.55007)] to deduce structural properties of \(TMF_1(3)^*(BSpin)\). Using these, he deduces that \(TMF_1(3)^*(BSpin)\) is determined by \(K_{\mathrm{Tate}}^*(BSpin)\) and rational information from the corresponding fact for \(TMF_1(3)^*\). Because the map \(TMF_1(3)^* \to K_{\mathrm{Tate}}^*\) is the \(q\)-expansion of modular forms, this is called a \(q\)-expansion principle; it can also be seen as a transchromatic statement because \(K_{\mathrm{Tate}}\) is of height \(1\) while \(TMF_1(3)\) is of height \(2\). Using the known K-theory of \(BSpin\), the author deduces the computation of \(TMF_1(3)^*(BSpin)\).NEWLINENEWLINEFor the case of \(BString\), the author considers instead \(\widehat{TMF}_1(3)\), which is defined to be \(K(2)\)-local \(TMF_1(3)\) and can be identified as a homotopy fixed point spectrum of Lubin-Tate theory \(E_2\). We have again that \(\widehat{TMF}_1(3)^*(BString)\) is a power series ring, but in addition to the Pontryagin classes we ha…
reviewed by: Lennart Meier
description: scientific article; zbMATH DE number 6559892