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 harv|Toda|1962.

Definition

See harv|Kochman|1990 or harv|Toda|1962 for more information.Suppose that:$W\; ightarrow^f\; X\; ightarrow^g\; Y\; ightarrow^h\; Z$is a sequence of maps between space, such that gf and hg are both nullhomotopic. Then we get a non-unique map from the coneCW 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 coneCX of X to Y from a homotopy from "hg" to a trivial map, which when composed with "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 ⟨f,g,h⟩ in the group [SW, Z] of homotopy classes of maps from the suspension SW to Z, called the Toda bracket off, g, and h. It 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 h [SW,Y] and [SX,Z] f.

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 stable for 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.harv|Nishida|1973.

If "f" and "g" and "h" are elements of π_∗^"S" with "f"⋅"g" = 0 and "g"⋅"h" = 0, there is a "Toda bracket" ⟨f,g,h⟩ 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. harvtxt|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.

References

*citation
last= Cohen|first= Joel M.
title= The decomposition of stable homotopy.
journal= Annals of Mathematics (2)
volume= 87
year= 1968
pages= 305–320
url= http://links.jstor.org/sici?sici=0003-486X%28196803%292%3A87%3A2%3C305%3ATDOSH%3E2.0.CO%3B2-K
doi=10.2307/1970586
id= MR|0231377 .
* citation
last= Kochman
first= Stanley O.
chapter=Toda brackets
pages=12-34
title= Stable homotopy groups of spheres. A computer-assisted approach
series= Lecture Notes in Mathematics
volume= 1423
publisher= Springer-Verlag
publication-place =Berlin
year= 1990
isbn= 978-3-540-52468-7
id= MR|1052407
doi= 10.1007/BFb0083797 .
* citation
last= Nishida
first= Goro
title= The nilpotency of elements of the stable homotopy groups of spheres
journal=Journal of the Mathematical Society of Japan
volume= 25
year= 1973
pages= 707–732
issn= 0025-5645
id= MR|0341485 .
* citation
last= Toda
first= Hirosi
title= Composition methods in homotopy groups of spheres
publisher= Princeton University Press
year= 1962
isbn= 978-0-691-09586-8
id=MR|0143217
series=Annals of Mathematics Studies
volume=49 .