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

M. Behrens, K. Ormsby, N. Stapleton, V. Stojanoska

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

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

Original languageEnglish
Pages (from-to)577-657
Number of pages81
JournalJournal of Topology
Volume12
Issue number2
DOIs
StatePublished - Jun 2019

Bibliographical note

Funding Information:
Received 23 January 2015; revised 27 January 2018; published online 6 March 2019. 2010 Mathematics Subject Classification 55N34, 55S25 (primary), 11F11, 11F33, 55T15 (secondary). The first author was partially supported by NSF CAREER grant DMS-1050466 and NSF grant DMS-1611786, the second author was partially supported by NSF Postdoctoral Fellowship DMS-1103873 and NSF grant DMS-1406327, the third author was partially supported by NSF grant DMS-0943787, and the fourth author was partially supported by NSF grant DMS-1307390/1606479.

Publisher Copyright:
© 2019 London Mathematical Society

ASJC Scopus subject areas

  • Geometry and Topology

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