#### Volume 17, issue 5 (2017)

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Higher Toda brackets and the Adams spectral sequence in triangulated categories

### J Daniel Christensen and Martin Frankland

Algebraic & Geometric Topology 17 (2017) 2687–2735
##### 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 ${d}_{r}$ in any Adams spectral sequence can be expressed as an $\left(r+1\right)$–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.

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##### Keywords
triangulated category, Adams spectral sequence, Toda bracket, cohomology operation, differential, higher order operation, projective class
Primary: 55T15
Secondary: 18E30
##### Publication
Received: 31 October 2015
Revised: 29 June 2016
Accepted: 22 February 2017
Published: 19 September 2017
##### Authors
 J Daniel Christensen Department of Mathematics University of Western Ontario London, ON Canada Martin Frankland Institut für Mathematik Universität Osnabrück Osnabrück Germany