Toda bracket

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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 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 ( 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.

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 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[1] as

for some b. This b is (a choice of) the Toda bracket in the group .

Convergence theorem

There is a convergence theorem originally due to Moss [1] which states that special Massey products 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 [2] 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 .

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  1. Moss, R. Michael F. (1970-08-01). "Secondary compositions and the Adams spectral sequence". Mathematische Zeitschrift. 115 (4): 283–310. doi:10.1007/BF01129978. ISSN   1432-1823. S2CID   122909581.
  2. Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli (2020-06-17). "More stable stems". arXiv: 2001.04511 [math.AT].