nLab
Toda bracket

Idea

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

$$\begin{array}{ccc}A& \to & B\\ \downarrow & & \downarrow & \searrow \\ 0& \to & B/(A)& \to & C\end{array}$$\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: $A\to B\to C\sim *:A\to C$ but now there is a choice to make, roughly parametrized by an action of ${\mathrm{Map}}_{*}(\Sigma A,C)$. 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 $A_0\to A_1\to A_2\to A_3$. If the composites $A_0\to A_2$ and $A_1\to A_3$ are nulhomotopic, then one has a diagram

$$\begin{array}{ccccc}{A}_0& \to & A_1& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ *& \to & A_2& \to & A_3\end{array}$$\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 $\Sigma A_0\to A_3$.

Preliminaries

Define $C$ and $D$ to be the cofibers of $A_0\to A_1$ and $A_1\to A_2$, respectively. A choice of homotopy $A_0\to A_2\sim 0$ corresponds to a choice of factorization $A_1\to C\to A_2$, which gives a diagram of pushout squares

$$\begin{array}{ccccc}{A}_0& \to & A_1& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ *& \to & C& \to & \Sigma A_0& \to & *\\ & & \downarrow & & \downarrow & & \downarrow \\ & & A_2& \to & D& \to & C\prime \end{array}$$\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 $\Sigma A_0\to D$ and possibly the object $C\prime$ depend on the choice of factor $C\to A_2$, but that $A_2\to D$ does not, in any meaningful sense, so depend: this is just the structure map of the cofiber of $A_1\to A_2$. Note that the cofiber $C\prime$ of $C\to A_2$ is thus equivalent to that of $\Sigma A_0\to D$; but again the role of choices must be studied.

Definitions

A sequence of maps $A_0\to A_1\to \cdots \to A_n$ will be called a bracket sequence (a novel phrase for the purposes of this entry) in either of two cases:

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

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

$${\Sigma }^m{A}_0\to D_m\to A_{m+2}\to A_{m+3}$$\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

$${\Sigma }^{m+1}{A}_0\to A_{m+3}.$$\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

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

If the Toda bracket for a bracket sequence includes the trivial map ${\Sigma }^{m+1}{A}_0\to A_{m+3}$ then the sequence is null-bracket.

Applications

By definition, if a sequence is a bracket sequence AND NOT a null-bracket sequence, it follows that all the relevant maps ${\Sigma }^k{A}_0\to A_n$ 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 ${\pi }_k{𝕊}^n$ for $k<31$ 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 ${\pi }_{*}𝕊$, and furthermore the maps in the bracket sequences can be chosen from a very small set (_FIXME_: be more precise! degree maps $n\iota$, Hopf map?s $\eta ,\theta ,\sigma$, and ${\alpha }_p$… )

Related concepts

• Massey product

References

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

• Hans-Joachim Baues, On the cohomology of categories, universal Toda brackets and homotopy pairs, K-Theory 11:3, April 1997, pp. 259-285 (27) springer
• Boryana Dimitrova, Universal Toda brackets of commutative ring spectra, poster, Bonn 2010, pdf
• C. Roitzheim, S. Whitehouse, Uniqueness of $A_{\mathrm{\infty }}$-structures and Hochschild cohomology, arxiv/0909.3222
• Steffen Sagave, Universal Toda brackets of ring spectra, Trans. Amer. Math. Soc., 360(5):2767-2808, 2008, math.KT/0611808