Toda bracket

In mathematics, the Toda bracket is an operation on homotopy classes of maps, in particular on homotopy groups of spheres, named after Hiroshi Toda who defined them and used them to compute homotopy groups of spheres in (Toda 1962).

Definition

See (Kochman 1990) or (Toda 1962) for more information. Suppose that

$W\stackrel{f}{\ \to\ } X\stackrel{g}{\ \to\ } Y\stackrel{h}{\ \to\ } Z$

is a sequence of maps between spaces, such that gf and hg are both nullhomotopic. Given a space A, let CA denote the cone of A. Then we get a non-unique map from CW to Y from a homotopy from gf to a trivial map, which when composed with h gives a map from CW to Z. Similarly we get a non-unique map from CX to Z from a homotopy from hg to a trivial map, which when composed with Cf, the cone of the map f, gives another map from CW to Z. By joining together these two cones on W and the maps from them to Z, we get a map 〈f, g, h〉 in the group [SW, Z] of homotopy classes of maps from the suspension SW to Z, called the Toda bracket of f, g, and h. It is not uniquely defined up to homotopy, because there was some choice in choosing the maps from the cones. Changing these maps changes the Toda bracket by adding elements of h[SW,Y] and [SX,Z]f.

There are also higher Toda brackets of several elements, defined when suitable lower Toda brackets vanish. This parallels the theory of Massey products in cohomology.

The Toda bracket for stable homotopy groups of spheres

The direct sum

$\pi_{\ast}^S=\bigoplus_{k\ge 0}\pi_k^S$

of the stable homotopy groups of spheres is a supercommutative graded ring, where multiplication (called composition product) is given by composition of representing maps, and any element of non-zero degree is nilpotent (Nishida 1973).

If f and g and h are elements of π_∗^S with f ⋅ g = 0 and g ⋅ h = 0, there is a Toda bracket 〈f, g, h〉 of these elements. The Toda bracket is not quite an element of a stable homotopy group, because it is only defined up to addition of composition products of certain other elements. Hiroshi Toda used the composition product and Toda brackets to label many of the elements of homotopy groups. Cohen (1968) showed that every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements.

The Toda bracket for general triangulated categories

In the case of a general triangulated category the Toda bracket can be defined as follows. Again, suppose that

$W\stackrel{f}{\ \to\ } X\stackrel{g}{\ \to\ } Y\stackrel{h}{\ \to\ } Z$

is a sequence of morphism in a triangulated category such that gf = 0 and hg = 0. Let $C_f$ denote the cone of f so we obtain an exact triangle

$W\stackrel{f}{\ \to\ } X\stackrel{i}{\ \to\ } C_f \stackrel{q}{\ \to\ } W[1]$

The relation gf = 0 implies that g factors (non-uniquely) through $C_f$ as

$X\stackrel{i}{\ \to\ } C_f \stackrel{a}{\ \to\ } Y$

for some a. Then, the relation hai = hg = 0 implies that ha factors (non-uniquely) through W[1] as

$C_f \stackrel{q}{\ \to\ } W[1] \stackrel{b}{\ \to\ } Z$

for some b. This b is (a choice of) the Toda bracket 〈f, g, h〉in the group $hom(W[1], Z)$.

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