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}
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