Whitehead product

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 (Whitehead 1941).

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

Definition

Given elements f ∈ π k ( X ) , g ∈ π l ( X ) {\displaystyle f\in \pi _{k}(X),g\in \pi _{l}(X)} , the Whitehead bracket

[ f , g ] ∈ π k + l − 1 ( X ) {\displaystyle [f,g]\in \pi _{k+l-1}(X)}

is defined as follows:

The product S k × S l {\displaystyle S^{k}\times S^{l}} can be obtained by attaching a ( k + l ) {\displaystyle (k+l)} -cell to the wedge sum

S k ∨ S l {\displaystyle S^{k}\vee S^{l}} ;

the attaching map is a map

S k + l − 1   ⟶   ϕ S k ∨ S l . {\displaystyle S^{k+l-1}{\stackrel {\phi }{\ \longrightarrow \ }}S^{k}\vee S^{l}.}

Represent f {\displaystyle f} and g {\displaystyle g} by maps

f : S k → X {\displaystyle f\colon S^{k}\to X}

and

g : S l → X , {\displaystyle g\colon S^{l}\to X,}

then compose their wedge with the attaching map, as

S k + l − 1   ⟶   ϕ S k ∨ S l   ⟶   f ∨ g X . {\displaystyle 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

π k + l − 1 ( X ) . {\displaystyle \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 π k ( X ) {\displaystyle \pi _{k}(X)} has degree ( k − 1 ) {\displaystyle (k-1)} ; equivalently, L k = π k + 1 ( X ) {\displaystyle L_{k}=\pi _{k+1}(X)} (setting L to be the graded quasi-Lie algebra). Thus L 0 = π 1 ( X ) {\displaystyle 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 ] {\displaystyle [f,g+h]=[f,g]+[f,h],[f+g,h]=[f,h]+[g,h]}
  • Graded Symmetry. [ f , g ] = ( − 1 ) p q [ g , f ] , f ∈ π p X , g ∈ π q X , p , q ≥ 2 {\displaystyle [f,g]=(-1)^{pq}[g,f],f\in \pi _{p}X,g\in \pi _{q}X,p,q\geq 2}
  • Graded Jacobi identity. ( − 1 ) p r [ [ f , g ] , h ] + ( − 1 ) p q [ [ g , h ] , f ] + ( − 1 ) r q [ [ h , f ] , g ] = 0 , f ∈ π p X , g ∈ π q X , h ∈ π r X  with  p , q , r ≥ 2 {\displaystyle (-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{\text{ 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 Uehara & Massey (1957) via the Massey triple product.

Relation to the action of π 1 {\displaystyle \pi _{1}}

If f ∈ π 1 ( X ) {\displaystyle f\in \pi _{1}(X)} , then the Whitehead bracket is related to the usual action of π 1 {\displaystyle \pi _{1}} on π k {\displaystyle \pi _{k}} by

[ f , g ] = g f − g , {\displaystyle [f,g]=g^{f}-g,}

where g f {\displaystyle g^{f}} denotes the conjugation of g {\displaystyle g} by f {\displaystyle f} .

For k = 1 {\displaystyle k=1} , this reduces to

[ f , g ] = f g f − 1 g − 1 , {\displaystyle [f,g]=fgf^{-1}g^{-1},}

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

Whitehead products on H-spaces

For a path connected H-space, all the Whitehead products on π ∗ ( X ) {\displaystyle \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 α ∈ π i ( X ) {\displaystyle \alpha \in \pi _{i}(X)} , β ∈ π j ( X ) {\displaystyle \beta \in \pi _{j}(X)} lie in the kernel of the suspension homomorphism Σ : π i + j − 1 ( X ) → π i + j ( Σ X ) {\displaystyle \Sigma \colon \pi _{i+j-1}(X)\to \pi _{i+j}(\Sigma X)}

Examples

  • [ i d S 2 , i d S 2 ] = 2 ⋅ η ∈ π 3 ( S 2 ) {\displaystyle [\mathrm {id} _{S^{2}},\mathrm {id} _{S^{2}}]=2\cdot \eta \in \pi _{3}(S^{2})} , where η : S 3 → S 2 {\displaystyle \eta \colon S^{3}\to S^{2}} is the Hopf map.

This can be shown by observing that the Hopf invariant defines an isomorphism π 3 ( S 2 ) ≅ Z {\displaystyle \pi _{3}(S^{2})\cong \mathbb {Z} } and explicitly calculating the cohomology ring of the cofibre of a map representing [ i d S 2 , i d S 2 ] {\displaystyle [\mathrm {id} _{S^{2}},\mathrm {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

References