\(K(1)\)-local topological modular forms
In 1995 Hopkins and Miller set up a diagram of elliptic spectra which can be thought of as a topological model for the stack of generalized elliptic curves. Its homotopy limit \(tmf\) is the spectrum of topological modular forms. The present paper deals with \(tmf\) in the \(K(1)\)-local category. In this category we have a fiber sequence \[ \begin{tikzcd}[column sep=2cm]E\to KO\wedge E\ar[r,"(\psi^3- 1)\wedge 1"] & KO\wedge E\rlap{\,.}\end{tikzcd} \] If \(E= S^0\) we obtain a map \(\pi_0KO\to \pi_{-1} S^0\). Let \(\xi\) be the image of the unit 1, and consider the homotopy pushouts \[ \begin{tikzcd} TS^{-1}\ar[r]\ar[d,"\xi " '] & T^*= S^0\ar[d]\\S^0\ar[r] & T_\xi\end{tikzcd}\qquad \begin{tikzcd} TS^0 \ar[r]\ar[d] &T*\ar[d]\\ T_\xi\ar[r] & M\end{tikzcd} \] in the category of \(K(1)\)-local \(E_\infty\) spectra. Here \(TX\) is the free \(E_\infty\) spectrum generated by the based space \(X\). The spectrum \(M\) can be considered as a \(K(1)\)-local model of \(tmf\). Using this model the author constructs the Witten orientation \(W: MO\langle 8\rangle\to tmf\): In the category of \(K(1)\)-local \(E_\infty\) spectra there is a splitting \(MO\langle 8\rangle\cong T_\xi\wedge\bigwedge^\infty_{i=1} TS^0\). There are classes \(x_0, x_1,\dots\in\pi_0MO\langle 8\rangle\) inducing an \(E_\infty\) map \(\bigwedge^\infty_{i=1} TS^0\to MO\langle 8\rangle\), and \(W\) is part of a homotopy pushout of \(E_\infty\) spectra \[ \begin{tikzcd} \bigwedge^\infty_{i=1} TS^0 \ar[r]\ar[d] & T*\ar[d]\\ MO\langle 8\rangle\ar[r,"W" '] & tmf \rlap{\,.}\end{tikzcd} \] The second main result of the paper is the calculation of the \(KO\)-homology of \(tmf\): Let \(D\) be the ring of divided congruences and \(SR\) the Moore spectrum associated with the ring \(R\), let \(L_{S\mathbb{Z}/2}\) denote the localization functor. Then there is a homotopy equivalence of ring spectra \[ L_{S\mathbb{Z}/2}(KO\wedge SD)\simeq L_{S\mathbb{Z}/2}(KO\wedge tmf). \] In particular, \(\pi_0(KO\wedge tmf)\cong D\) after 2-completion. The paper starts with a recollection of basic facts about modular forms, divided congruences, and elliptic cohomology theories. Then \(E_\infty\) elliptic spectra are considered, the 2-cell model of \(tmf\) is constructed, and the proofs of the main results are given.
scientific article; zbMATH DE number 2130986
scientific article