Algebraic & Geometric Topology

Higher Toda brackets and the Adams spectral sequence in triangulated categories

J Daniel Christensen and Martin Frankland

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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 dr in any Adams spectral sequence can be expressed as an (r+1)–fold Toda bracket and as an rth 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.

Article information

Algebr. Geom. Topol., Volume 17, Number 5 (2017), 2687-2735.

Received: 31 October 2015
Revised: 29 June 2016
Accepted: 22 February 2017
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55T15: Adams spectral sequences
Secondary: 18E30: Derived categories, triangulated categories

triangulated category Adams spectral sequence Toda bracket cohomology operation differential higher order operation projective class


Christensen, J Daniel; Frankland, Martin. Higher Toda brackets and the Adams spectral sequence in triangulated categories. Algebr. Geom. Topol. 17 (2017), no. 5, 2687--2735. doi:10.2140/agt.2017.17.2687.

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