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

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.

v4: Minor changes made in response to referee report. To appear in J. Topology. 87 pages, 19 figures

References in corpus (3)

• On the homotopy of Q(3) and Q(5) at the prime 2
• Connective versions of TMF(3)
• A class of $2$-local finite spectra which admit a $v_2^1$-self-map

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• On Lurie's theorem and applications
• The $BP⟨2⟩$-cooperations at odd primes
• ${\mathcal{E}}_{\infty }$ ring spectra and elements of Hopf invariant $1$