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 $d_r$ 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 three-fold Toda brackets in principle determine the higher Toda brackets.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2015
- arXiv:
- arXiv:1510.09216
- Bibcode:
- 2015arXiv151009216C
- Keywords:
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- Mathematics - Algebraic Topology;
- Mathematics - Category Theory;
- 55T15 (Primary) 18E30;
- 55S20;
- 18G25 (Secondary)
- E-Print:
- v2: Added Section 7, about an application to computing maps between modules over certain ring spectra. Minor improvements elsewhere. v3: Minor updates throughout