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

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

doi:10.1112/topo.12094

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

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

Mathematics

Research output: Contribution to journal › Article › peer-review

8 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 language	English
Pages (from-to)	577-657
Number of pages	81
Journal	Journal of Topology
Volume	12
Issue number	2
DOIs	https://doi.org/10.1112/topo.12094
State	Published - Jun 2019

Bibliographical note

Publisher Copyright:
© 2019 London Mathematical Society

Funding

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.

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	DMS-1050466, DMS-1611786, DMS-0943787, DMS-1307390/1606479, DMS-1103873, DMS-1406327
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China
Directorate for Mathematical and Physical Sciences	1050466, 1307390, 0943787, 1103873
Directorate for Mathematical and Physical Sciences

ASJC Scopus subject areas

Geometry and Topology

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10.1112/topo.12094

Cite this

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title = "On the ring of cooperations for 2-primary connective topological modular forms",

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{\textendash}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{\textendash}Mahowald{\textendash}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.",

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AB - 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.

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

Abstract

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