Higher Toda brackets and the Adams spectral sequence in triangulated categories

Autor(en): Christensen, J. Daniel
Frankland, Martin
Stichwörter: FUNCTORS; GHOSTS; Mathematics; MODEL; MODULES; STABLE-HOMOTOPY
Erscheinungsdatum: 2017
Herausgeber: GEOMETRY & TOPOLOGY PUBLICATIONS
Journal: ALGEBRAIC AND GEOMETRIC TOPOLOGY
Volumen: 17
Ausgabe: 5
Startseite: 2687
Seitenende: 2735
Zusammenfassung: 
The Adams spectral sequence is available in any triangulated category equipped with a projective or injective class. Higher Toda brackets can also be defined in a triangulated category, as observed by B Shipley based on J Cohen's approach for spectra. We provide a family of definitions of higher Toda brackets, show that they are equivalent to Shipley's and show that they are self-dual. Our main result is that the Adams differential d(r) in any Adams spectral sequence can be expressed as an (r 1)-fold Toda bracket and as an r th order cohomology operation. We also show how the result simplifies under a sparseness assumption, discuss several examples and give an elementary proof of a result of Heller, which implies that the 3-fold Toda brackets in principle determine the higher Toda brackets.
ISSN: 14722739
DOI: 10.2140/agt.2017.17.2687

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