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 selfdual. 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 $r^{\text{th}}$ 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 threefold Toda brackets in principle determine the higher Toda brackets.
 Publication:

arXiv eprints
 Pub Date:
 October 2015
 arXiv:
 arXiv:1510.09216
 Bibcode:
 2015arXiv151009216C
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Category Theory;
 55T15 (Primary) 18E30;
 55S20;
 18G25 (Secondary)
 EPrint:
 v2: Added Section 7, about an application to computing maps between modules over certain ring spectra. Minor improvements elsewhere. v3: Minor updates throughout