Higher Toda brackets and the Adams spectral sequence in triangulated categories

TL;DR

This paper explores the relationship between higher Toda brackets and the Adams spectral sequence in triangulated categories, providing new definitions, equivalences, and expressing differentials as Toda brackets and cohomology operations.

Contribution

It introduces a family of equivalent definitions of higher Toda brackets, proves their self-duality, and expresses Adams differentials as Toda brackets and higher cohomology operations.

Findings

Adams differentials can be expressed as higher Toda brackets.

Higher Toda brackets are equivalent to Shipley's definitions.

Simplifications occur under sparseness assumptions.

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.