Higher Toda brackets and the Adams spectral sequence in triangulated categories

Christensen, J Daniel; Frankland, Martin

doi:10.2140/agt.2017.17.2687

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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

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 $d_{r}$ in any Adams spectral sequence can be expressed as an $(r + 1)$ –fold Toda bracket and as an $r^{t h}$ 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.

Keywords

triangulated category, Adams spectral sequence, Toda bracket, cohomology operation, differential, higher order operation, projective class

Mathematical Subject Classification 2010

Primary: 55T15

Secondary: 18E30

References

Publication

Received: 31 October 2015

Revised: 29 June 2016

Accepted: 22 February 2017

Published: 19 September 2017

Authors

J Daniel Christensen
Department of Mathematics University of Western Ontario London, ON Canada

Martin Frankland
Institut für Mathematik Universität Osnabrück Osnabrück Germany

Volume 17, issue 5 (2017)