Whitehead product explained

In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in .

The relevant MSC code is: 55Q15, Whitehead products and generalizations.

Definition

Given elements

f \in \pi _{k(X),} g \in \pi _{l(X)} , the Whitehead bracket [f,g] \in \pi _{k+l-1} (X)

is defined as follows:

The product

S ^{k} x S ^{l} can be obtained by attaching a (k + l) -cell to the wedge sumS ^{k} \vee S ^{l}

the attaching map is a map

S ^{k+l-1} \stackrel{\phi}{ \longrightarrow } S ^{k} \vee S ^{l.}

Represent

f and g by mapsf\colon S ^{k} \to X

and

g\colon S ^{l} \to X,

then compose their wedge with the attaching map, as

S ^{k+l-1} \stackrel{\phi}{ \longrightarrow } S ^{k} \vee S ^{l} \stackrel{f \vee g}{ \longrightarrow } X .

The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of

\pi _{k+l-1} (X).

Grading

Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so

\pi _{k(X)} has degree (k -1 ) ; equivalently, L _{k} = \pi _{k+1} (X) (setting L to be the graded quasi-Lie algebra). Thus L _{0} = \pi _{1(X)} acts on each graded component.

Properties

The Whitehead product satisfies the following properties:

• Bilinearity.

[f,g + h] = [f,g] + [f,h], [f + g,h] = [f,h] + [g,h]

• Graded Symmetry.

[f,g] = ( -1 ) ^{pq} [g,f], f \in \pi _{p} X, g \in \pi _{q} X, p,q \geq 2

• Graded Jacobi identity.

( -1 ) ^{pr} [[f,g],h] + ( -1 ) ^{pq} [[g,h],f] + ( -1 ) ^{rq} [[h,f],g] = 0 , f \in \pi _{p} X, g \in \pi _{q} X, h \in \pi _{r} X with p,q,r \geq 2

Sometimes the homotopy groups of a space, together with the Whitehead product operation are called a graded quasi-Lie algebra; this is proven in via the Massey triple product.

Relation to the action of

\pi _{1}

If

f \in \pi _{1(X)} , then the Whitehead bracket is related to the usual action of \pi _{1} on \pi _{k} by[f,g] = g ^{f-g,}

where

g ^{f} denotes the conjugation of g by f .

For

k =1 , this reduces to[f,g] = fgf ^{-1} g ^{-1} ,

which is the usual commutator in

\pi _{1(X)} . This can also be seen by observing that the 2 -cell of the torus S ^{1} x S ^{1} is attached along the commutator in the 1 -skeleton S ^{1} \vee S ^{1} .

Whitehead products on H-spaces

For a path connected H-space, all the Whitehead products on

\pi _{*} (X) vanish. By the previous subsection, this is a generalization of both the facts that the fundamental groups of H-spaces are abelian,and that H-spaces are simple.

Suspension

All Whitehead products of classes

\alpha \in \pi _{i} (X) , \beta \in \pi _{j} (X) lie in the kernel of the suspension homomorphism \Sigma \colon \pi _{i+j-1} (X) \to \pi _{i+j} (\Sigma X)

Examples

[ id _{S ^{2}} , id _{S ^{2}} ] = 2 ⋅ η \in \pi ^{2} _{3 (S} ) , where η \colon S ^{3} \to S ^{2} is the Hopf map.

This can be shown by observing that the Hopf invariant defines an isomorphism

\pi _{3} (S ^{2} ) \cong \Z and explicitly calculating the cohomology ring of the cofibre of a map representing [ id _{S ^{2}} , id _{S ^{2}} ] . Using the Pontryagin–Thom construction there is a direct geometric argument, using the fact that the preimage of a regular point is a copy of the Hopf link.

See also

• Generalised Whitehead product
• Massey product
• Toda bracket

References

• • • • Book: Whitehead , George W. . George W. Whitehead

. George W. Whitehead. Elements of homotopy theory. X.7 The Whitehead Product. Springer-Verlag. 978-0387903361. 472–487. 1978.