On the ring of cooperations for 2-primary connective topological modular forms

TL;DR

This paper investigates the structure of the ring of cooperations for 2-primary connective topological modular forms (tmf), using spectral sequences, modular forms, and comparisons with TMF, providing new insights and explicit computations.

Contribution

It introduces multiple perspectives on tmf_*tmf, relates them, and computes low-degree parts, including new descriptions and connections to classical bo_*bo theory.

Findings

Decomposition of the Adams spectral sequence for tmf^tmf

Description of tmf_*tmf in terms of modular forms

Identification of parts of tmf^tmf as connective covers of TMF_0(3) and TMF_0(5)

Abstract

We analyze the ring tmf_*tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E_2-term of the Adams spectral sequence for tmf ^ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf_*tmf in the rationalization of TMF_*TMF admits a description in terms of 2-variable modular forms, and (3) modulo v_2-torsion, tmf_*tmf injects into a certain product of copies of TMF_0(N)_*, for various values of N. We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf_*tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ^ tmf gives a connective cover of TMF_0(3), and show that another piece gives a connective cover of TMF_0(5). To help motivate our methods, we also review the existing work on bo_*bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.