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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

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.

triangulated category, Adams spectral sequence, Toda bracket, cohomology operation, differential, higher order operation, projective class
Mathematical Subject Classification 2010
Primary: 55T15
Secondary: 18E30
Received: 31 October 2015
Revised: 29 June 2016
Accepted: 22 February 2017
Published: 19 September 2017
J Daniel Christensen
Department of Mathematics
University of Western Ontario
London, ON
Martin Frankland
Institut für Mathematik
Universität Osnabrück