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2017 Higher Toda brackets and the Adams spectral sequence in triangulated categories
J Daniel Christensen, Martin Frankland
Algebr. Geom. Topol. 17(5): 2687-2735 (2017). DOI: 10.2140/agt.2017.17.2687

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 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.

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J Daniel Christensen. Martin Frankland. "Higher Toda brackets and the Adams spectral sequence in triangulated categories." Algebr. Geom. Topol. 17 (5) 2687 - 2735, 2017. https://doi.org/10.2140/agt.2017.17.2687

Information

Received: 31 October 2015; Revised: 29 June 2016; Accepted: 22 February 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 06791380
MathSciNet: MR3704239
Digital Object Identifier: 10.2140/agt.2017.17.2687

Subjects:
Primary: 55T15
Secondary: 18E30

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.17 • No. 5 • 2017
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