On the ring of cooperations for 2‐primary connective topological modular forms. Issue 2 (6th March 2019) - Behrens, M.; Ormsby, K.; Stapleton, N.; Stojanoska, V.
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 TMF ∗ TMF Q 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 π ∗ 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.