## Algebraic & Geometric Topology

### Higher Toda brackets and the Adams spectral sequence in triangulated categories

#### Abstract

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

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

Dates
Revised: 29 June 2016
Accepted: 22 February 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841479

Digital Object Identifier
doi:10.2140/agt.2017.17.2687

Mathematical Reviews number (MathSciNet)
MR3704239

Zentralblatt MATH identifier
06791380

Subjects
Secondary: 18E30: Derived categories, triangulated categories

#### Citation

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. https://projecteuclid.org/euclid.agt/1510841479

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