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 .

Definition

See or for more information. Suppose that

is a sequence of maps between spaces, such that the compositions are both nullhomotopic denote the cone. Then we get a (non-unique) map

induced by a homotopy gives a map

1. .

, gives another map,

1. .

, we get a map

, called the Toda bracket.

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

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 .

If f and g and h, there is a Toda bracket 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. 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

is a sequence of morphism in a triangulated category denote the cone of f so we obtain an exact triangle

implies that g factors (non-uniquely) as

factors (non-uniquely) through W[1] as

for some b. This b.

Convergence theorem

There is a convergence theorem originally due to Moss-page of the Adams spectral sequence are permanent cyclespg 18-19.

References

• .

• .

• .

• .