Toda bracket explained
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
W\stackrel{f}{ \to }X\stackrel{g}{ \to }Y\stackrel{h}{ \to }Z
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
\langlef,g,h\rangle\colonSW\toZ
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 .
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. 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
W\stackrel{f}{ \to }X\stackrel{g}{ \to }Y\stackrel{h}{ \to }Z
is a sequence of morphism in a
triangulated category such that
and
. Let
denote the cone of
f so we obtain an exact triangle
W\stackrel{f}{ \to }X\stackrel{i}{ \to }Cf\stackrel{q}{ \to }W[1]
The relation
implies that
g factors (non-uniquely) through
as
X\stackrel{i}{ \to }Cf\stackrel{a}{ \to }Y
for some
. Then, the relation
implies that
factors (non-uniquely) through
W[1] as
Cf\stackrel{q}{ \to }W[1]\stackrel{b}{ \to }Z
for some
b. This
b is (a choice of) the Toda bracket
in the group
\operatorname{hom}(W[1],Z)
.
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
\langle\alpha,\beta,\gamma\rangle
in
for elements
lifting
.
References
Notes and References
- Moss. R. Michael F.. 1970-08-01. Secondary compositions and the Adams spectral sequence. Mathematische Zeitschrift. en. 115. 4. 283–310. 10.1007/BF01129978. 122909581. 1432-1823.
- Isaksen. Daniel C.. Wang. Guozhen. Xu. Zhouli. 2020-06-17. More stable stems. math.AT. 2001.04511.
