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 .
Definition.
See or for more information.
Suppose that
is a sequence of maps between spaces, such that the compositions formula_2 and formula_3 are both nullhomotopic. Given a space formula_4, let formula_5 denote the cone of formula_4. Then we get a (non-unique) map 
induced by a homotopy from formula_2 to a trivial map, which when post-composed with formula_9 gives a map 
Similarly we get a non-unique map formula_11 induced by a homotopy from formula_3 to a trivial map, which when composed with formula_13, the cone of the map formula_14, gives another map, 
By joining together these two cones on formula_16 and the maps from them to formula_17, we get a map 
representing an element in the group formula_19 of homotopy classes of maps from the suspension formula_20 to formula_17, called the Toda bracket of formula_14, formula_23, and formula_9. The map formula_25 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 formula_26 and formula_27.
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" are elements of formula_29 with formula_30 and formula_31, there is a "Toda bracket" formula_25 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 such that formula_34 and formula_35. Let formula_36 denote the cone of "f" so we obtain an exact triangle
The relation formula_34 implies that "g" factors (non-uniquely) through formula_36 as 
for some formula_41. Then, the relation formula_42 implies that formula_43 factors (non-uniquely) through "W[1]" as 
for some "b". This "b" is (a choice of) the Toda bracket formula_25 in the group formula_46.
Convergence theorem.
There is a convergence theorem originally due to Moss which states that special Massey products formula_47 of elements in the formula_48-page of the Adams spectral sequence contain a permanent cycle, meaning has an associated element in formula_49, assuming the elements formula_50 are permanent cyclespg 18-19. Moreover, these Massey products have a lift to a motivic Adams spectral sequence giving an element in the Toda bracket formula_51 in formula_52 for elements formula_53 lifting formula_50.