Constructing and calculating Adams operations on dualisable topological modular forms

TL;DR

This paper constructs and computes Adams operations on the topological modular forms cohomology theory, providing new tools for understanding its structure and applications in stable homotopy theory.

Contribution

It introduces the first stable Adams operations on Tmf and calculates their effects, enabling new constructions in the field.

Findings

Adams operations on Tmf are explicitly constructed.

Calculations on Tmf-cohomology of spheres are achieved.

Applications include new spectra analogues and insights into stable homotopy groups.

Abstract

We construct Adams operations on the cohomology theory Tmf of topological modular forms; the first such stable operations on this cohomology theory. These Adams operations are then calculated on the Tmf-cohomology of spheres 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.