# nLab Toda bracket

## Theorems

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## 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 $B/(A)$:

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

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 $Map_* (\Sigma A, C)$. 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

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

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

$\begin{array}{ccccccc} A_0 & \to & A_1 & \to & * \\ \downarrow & & \downarrow & & \downarrow \\ * & \to & C &\to & \Sigma A_0 & \to & *\\ & & \downarrow & & \downarrow & & \downarrow \\ & & A_2 & \to & D & \to & C' \end{array}$

It is to be noted that the map $\Sigma A_0 \to D$ and possibly the object $C'$ 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 $C'$ 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
• $n \gt 3$, 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

$\Sigma^m A_0 \to D_m \to A_{m+2} \to A_{m+3}$

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

$\Sigma^{m+1} A_0 \to A_{m+3} .$

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
• $n \gt 2$, 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 $\pi_k \mathbb{S}^n$ for $k \lt 31$ 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 $\pi_* \mathbb{S}$, and furthermore the maps in the bracket sequences can be chosen from a very small set (_FIXME_: be more precise! degree maps $n \iota$, Hopf maps $\eta, \theta,\sigma$, and $\alpha_p$… )

## References

• 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 (195.37.209.180)