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

Producción científica: Article › revisión exhaustiva

7 Citas (Scopus)

Resumen

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.

Idioma original	English
Páginas (desde-hasta)	577-657
Número de páginas	81
Publicación	Journal of Topology
Volumen	12
N.º	2
DOI	https://doi.org/10.1112/topo.12094
Estado	Published - jun 2019

Nota bibliográfica

Publisher Copyright:
© 2019 London Mathematical Society

Financiación

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.

Financiadores	Número del financiador
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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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–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

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