Characteristic classes in \(TMF\) of level \(\Gamma_{1}(3)\) (Q2796094)

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 have another generator \(r\), which is lifted from \(K(2)^*K(\mathbb{Z},3) \cong \mathbb{F}_p[v_2^{\pm 1}][[r]]\). The last isomorphism is computed as Theorem 12.4 in [\textit{D.C. Ravenel} and \textit{W.S. Wilson}, Am. J. Math. 102, 691--748 (1980; Zbl 0466.55007)] for odd primes in a way that also works for \(p=2\) as explained in the appendix of [\textit{D.C. Johnson} and \textit{W.S. Wilson}, Am. J. Math. 107, 427--453 (1985; Zbl 0574.55003)].NEWLINENEWLINEThe paper ends with an application to loop groups. More precisely, it constructs an additive map from an abelian group of positive energy representation of looped spin groups to \(\widehat{TMF}(n)^0(BString)\) for \(n\) divisible by \(3\).