Toda bracket


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



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(,1)(\infty,1)-topos theory

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos



In abelian categories one talks of chain complexes; and in that context a composable pair ABC A \to B \to C is null iff BC B \to C factors through the cokernel B/(A) B/(A) :

A B 0 B/(A) C \begin{array}{ccccc} A & \to & B \\ \downarrow & & \downarrow & \searrow \\ 0 & \to & B/(A) & \to & C \end{array}

and so forth. In a strict context, the factorization is unique.

In a pointed (∞,1)-category with (∞,1)-colimits of small 1-truncated diagrams, one may still consider factorizations through cofibers: ABC*:AC A \to B \to C \sim * : A \to C but now there is a choice to make, roughly parametrized by an action of Map *(ΣA,C)Map_* (\Sigma A, C). This leads to interesting structure, describing (with upper bounds!) how trivially a particular sequence of arrows may compose.

To begin, consider a sequence of maps A 0A 1A 2A 3 A_0 \to A_1 \to A_2 \to A_3 . If the composites A 0A 2A_0 \to A_2 and A 1A 3 A_1\to A_3 are nulhomotopic, then one has a diagram

A 0 A 1 * * A 2 A 3\begin{array}{ccccc} A_0 & \to & A_1 & \to & * \\ \downarrow & & \downarrow & & \downarrow \\ * & \to & A_2 & \to & A_3 \end{array}

any choice of homotopies in the two squares gives a map ΣA 0A 3 \Sigma A_0 \to A_3 .


Define CC and DD to be the cofibers of A 0A 1A_0 \to A_1 and A 1A 2A_1\to A_2, respectively. A choice of homotopy A 0A 20 A_0 \to A_2 \sim 0 corresponds to a choice of factorization A 1CA 2 A_1 \to C \to A_2 , which gives a diagram of pushout squares

A 0 A 1 * * C ΣA 0 * A 2 D C\begin{array}{ccccccc} A_0 & \to & A_1 & \to & * \\ \downarrow & & \downarrow & & \downarrow \\ * & \to & C &\to & \Sigma A_0 & \to & *\\ & & \downarrow & & \downarrow & & \downarrow \\ & & A_2 & \to & D & \to & C' \end{array}

It is to be noted that the map ΣA 0D \Sigma A_0 \to D and possibly the object CC' depend on the choice of factor CA 2 C \to A_2 , but that A 2DA_2 \to D does not, in any meaningful sense, so depend: this is just the structure map of the cofiber of A 1A 2A_1\to A_2. Note that the cofiber CC' of CA 2C\to A_2 is thus equivalent to that of ΣA 0D\Sigma A_0 \to D; but again the role of choices must be studied.


A sequence of maps A 0A 1A nA_0 \to A_1 \to \cdots \to A_n will be called a bracket sequence (a novel phrase for the purposes of this entry) in either of two cases:

  • n=3n = 3 and the composites A 0A 2A_0 \to A_2 and A 1A 3A_1 \to A_3 are nulhomotopic; OR
  • n>3n \gt 3, and (using the preceding notations), there are choices of factor CA 2C\to A_2 and DA 3 D \to A_3 such that the induced sequence ΣA 0DA 3A n \Sigma A_0 \to D \to A_3 \to \cdots \to A_n is a bracket sequence.

In all cases, a bracket sequence leads to a three-map sequence

Σ mA 0D mA m+2A m+3 \Sigma^m A_0 \to D_m \to A_{m+2} \to A_{m+3}

in which consecutive maps compose trivially, and so there are induced choices of maps

Σ m+1A 0A m+3. \Sigma^{m+1} A_0 \to A_{m+3} .

The collection of all such maps, taking all compatible variations, is the Toda Bracket of the bracket sequence.

Among the bracket sequences, a particular family arises which here will be called null-bracket (again, a novel phrase). A sequence will be called null-bracket if

  • n=2n=2 and A 0A 2A_0 \to A_2 is trivial, OR
  • n>2n \gt 2, and there is a choice of factorization A 1CA 2 A_1 \to C \to A_2 such that the sequence CA 2A n C \to A_2 \to \cdots \to A_n is null-bracket.

If the Toda bracket for a bracket sequence includes the trivial map Σ m+1A 0A m+3\Sigma^{m+1} A_0 \to A_{m+3} then the sequence is null-bracket.


By definition, if a sequence is a bracket sequence AND NOT a null-bracket sequence, it follows that all the relevant maps Σ kA 0A n\Sigma^{k} A_0 \to A_n are nontrivial. Things like these Toda brackets have been studied by many (FIXME: referrences later) and especially the length-three brackets used by H. Toda to describe most of π k𝕊 n\pi_k \mathbb{S}^n for k<31k \lt 31 or so.

In (Cohen, 1968) is given a criterion for stable maps of spheres to inhabit non-null Toda brackets; this turns out to be most of π *𝕊\pi_* \mathbb{S}, and furthermore the maps in the bracket sequences can be chosen from a very small set (_FIXME_: be more precise! degree maps nιn \iota, Hopf maps η,θ,σ\eta, \theta,\sigma, and α p\alpha_p… )


  • Joel Cohen, The decomposition of stable homotopy, Annals of Mathematics (2) 87 (2): 305–320 (1968)

Revised on February 17, 2016 12:43:01 by Urs Schreiber (