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 ).
See ( Kochman 1990 ) or ( Toda 1962 ) for more information. Suppose that
is a sequence of maps between spaces, such that the compositions and are both nullhomotopic. Given a space , let denote the cone of . Then we get a (non-unique) map
induced by a homotopy from to a trivial map, which when post-composed with gives a map
Similarly we get a non-unique map induced by a homotopy from to a trivial map, which when composed with , the cone of the map , gives another map,
By joining these two cones on and the maps from them to , we get a map
representing an element in the group of homotopy classes of maps from the suspension to , called the Toda bracket of , , and . The map 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 and .
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 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 ( Nishida 1973 ).
If f and g and h are elements of with and , 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. 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.
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 and . Let denote the cone of f so we obtain an exact triangle
The relation implies that g factors (non-uniquely) through as
for some . Then, the relation implies that factors (non-uniquely) through W as
for some b. This b is (a choice of) the Toda bracket in the group .
There is a convergence theorem originally due to Moss of elements in the -page of the Adams spectral sequence contain a permanent cycle, meaning has an associated element in , assuming the elements are permanent cycles pg 18-19. Moreover, these Massey products have a lift to a motivic Adams spectral sequence giving an element in the Toda bracket in for elements lifting .which states that special Massey products
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