High Quality Content by WIKIPEDIA articles! 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 Wstackrel{f}{ to } Xstackrel{g}{ to } Ystackrel{h}{ to } Z is a sequence of maps between space, such that gf and hg are both nullhomotopic. Then we get a non-unique map from the cone CW of W 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 the cone CX of X 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 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.

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