Constructing and calculating Adams operations on dualisable topological modular forms

Abstract

We construct Adams operations ${\psi }^k$ on the cohomology theory $\mathrm{Tmf}$ of dualisable topological modular forms after inverting $k$; the first such multiplicative stable operations on this cohomology theory. These Adams operations are then calculated on the homotopy groups of $\mathrm{Tmf}$ using a combination of descent spectral sequences and Anderson duality. Applications of these operations are then given, including constructions of connective height $2$ analogues of Adams summands and image-of-$J$ spectra.