Note on Toda brackets
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Note on Toda brackets Samik Basu1 · David Blanc2 · Debasis Sen3 Received: 4 November 2019 / Accepted: 13 August 2020 © Tbilisi Centre for Mathematical Sciences 2020
Abstract We provide a general definition of Toda brackets in a pointed model category, show how they serve as obstructions to rectification, and explain their relation to the classical stable operations. Keywords Higher homotopy operations · Toda brackets · Stable homotopy Mathematics Subject Classification Primary 55Q35; Secondary 55P99 · 55Q40
Introduction Toda brackets, defined by Toda in [19], play an important role in homotopy theory both for their original purpose of calculating homotopy groups in [20], and because they serve as differentials in spectral sequences (see [1,3,11]). The notion has been generalized in several ways (see, e.g., [4,9,21]), not all of which agree. In this note we provide a definition of higher Toda brackets in a general pointed model category C, show how these appear as the successive obstructions to strictifying certain diagrams (namely, chain complexes in the homotopy category ho C) —see Theorem 1.23 below. In Sect. 2, we explain the connection with the traditional stable
Communicated by Tim Porter.
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Debasis Sen [email protected] Samik Basu [email protected] David Blanc [email protected]
1
Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, India
2
Department of Mathematics, University of Haifa, 3498838 Haifa, Israel
3
Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, Uttar Pradesh 208016, India
123
S. Basu et al.
description in terms of filtered complexes. In Sect. 3, we provide two examples of settings in which higher Toda brackets occur.
1 Higher Toda brackets We provide a variant of the definition of higher Toda brackets sketched in [9, §7] which brings out clearly their connection to rectification of linear diagrams. Definition 1.1 Let C be a cofibrantly generated left proper pointed model category. A Toda diagram of length n for C is a diagram in the homotopy category ho C of the form [ f0 ]
[ f n−1 ]
[ f1 ]
A0 −−→ A1 −−→ A2 → · · · → An−1 −−−→ An
(1.2)
with [ f k ] ◦ [ f k−1 ] = 0 for each 1 ≤ k < n. A strictification of (1.2) is a diagram in C of the form f 0
f n−1
f 1
→ A1 − → A2 → · · · → An−1 −−→ An A0 −
(1.3)
= 0 (the strict zero map) for each 1 ≤ k < n, which is weakly with f k ◦ f k−1 equivalent to a lift of (1.2).
To obtain a homotopy meaningful description of such strictifications, we shall need: Definition 1.4 Let I2n denote the lattice of subsets of [n] = {1, 2, . . . , n}, which we shall think of as an n-dimensional cube with vertices labelled by J = (ε1 , ε2 , . . . , εn ) with each εi ∈ {0, 1}, so J is the characteristic function of a subset of [n]. The cube is partially ordered in the usual way, with a unique map J → J whenever εi ≤ εi for all 1 ≤ i ≤ n. Write Jk for the vertex labelled by k ones followed by n − k zeros (k = 0, 1, . . . , n). Note that any strictification (1.3) extends uniquely to a diag
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