Connective models for topological modular forms of level \(n\) (Q6091922)

The \(\mathbb{E}_{\infty}\)-ring spectrum of topological modular forms is an enhancement of the ring of holomorphic modular forms to the setting of brave new algebra. Using a connective version of this \(\mathbb{E}_{\infty}\)-ring spectrum, denoted \(\mathrm{tmf}\), one can lift the Witten genus to a map of \(\mathbb{E}_{\infty}\)-ring spectra \(\mathrm{MString}\to \mathrm{tmf}\). The paper under review studies connective models for topological modular forms with respect to certain congruence subgroups of \(\mathrm{SL}_2(\mathbb{Z})\) such as \(\Gamma=\Gamma_0(n)\), \(\Gamma_1(n)\), and \(\Gamma(n)\). The obstruction to building nice models for connective versions of topological modular forms in these cases lives in the fundamental group of these \(\mathbb{E}_{\infty}\)-ring spectra and the author therefore produces a general formalism for killing \(\pi_1\) in an \(\mathbb{E}_{\infty}\)-ring spectrum based on ideas of Lawson. As a result the author produces nice connective \(\mathbb{E}_{\infty}\)-ring spectra \(\mathrm{tmf}_1(n)\) that recover the ring of homolomorphic forms with respect to \(\Gamma_1(n)\). The author then produces complex orientations \(\mathrm{MU}\to \mathrm{tmf}_1(n)\) realizing the Hirzebruch level-\(n\)-genus. The author also constructs \(C_2\)-equivariant refinements of \(\mathrm{tmf}_1(n)\) and a \(C_2\)-equivariant refinement \(\mathrm{MU}_{\mathbb{R}}\to \mathrm{tmf}_1(n)\) of these complex orientations. The author goes on to prove several nice properties of these \(\mathbb{E}_{\infty}\)-ring spectra, for example \(\mathrm{tmf}_0(n)\), \(\mathrm{tmf}_1(n)\) and \(\mathrm{tmf}(n)\) are perfect \(\mathrm{tmf}[1/n]\)-modules and therefore they have finitely presented mod \(p\)-cohomology as modules over the Steenrod algebra. This condition has proven useful for proving Lichtenbaum-Quillen type results in algebraic \(K\)-theory of ring spectra following a program of Ausoni and Rognes. Finally, for \(n>1\) odd, the author proves that the \(C_2\)-equivariant \(\mathrm{tmf}_1(n)_{(2)}\) splits as a wedge of \(\rho\)-divisible suspensions of \(\mathrm{tmf}_1(3)_{(2)}\) where \(\rho\) is the regular representation of \(C_2\). This kind of condition is useful in the study of evenness in \(C_2\)-equivariant homotopy theory. Overall the paper is well written, clear, and enjoyable to read.