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

摘要

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 E2-term of the Adams spectral sequence for tmf perpendicular to tmf admits a decomposition in terms of Ext groups for bo -Brown-Gitler modules, (2) the image of tmf * tmf in TMF * TMF Q admits a description in terms of 2-variable modular forms, and (3) modulo v2-torsion, tmf * tmf injects into a certain product of copies of pi* 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 perpendicular to 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.