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| In mathematics, the '''Whitehead product''' is a [[Graded Lie algebra|graded]] [[quasi-Lie algebra]] structure on the [[homotopy group]]s of a space. It was defined by [[J. H. C. Whitehead]] in {{harv|Whitehead|1941}}.
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| == Definition ==
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| Given elements <math>f \in \pi_k(X), g \in \pi_l(X)</math>, the '''Whitehead bracket'''
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| :<math>[f,g] \in \pi_{k+l-1}(X) \, </math>
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| is defined as follows:
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| The product <math>S^k \times S^l</math> can be obtained by attaching a <math>(k+l)</math>-cell to the [[wedge sum]] | |
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| :<math>S^k \vee S^l</math>;
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| the [[attaching map]] is a map
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| :<math>S^{k+l-1} \to S^k \vee S^l. \,</math>
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| Represent <math>f</math> and <math>g</math> by maps
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| :<math>f\colon S^k \to X \, </math>
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| and
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| :<math>g\colon S^l \to X, \, </math>
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| then compose their wedge with the attaching map, as
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| :<math>S^{k+l-1} \to S^k \vee S^l \to X \, </math>
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| The [[homotopy class]] of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of
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| :<math>\pi_{k+l-1}(X). \, </math>
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| ==Grading==
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| Note that there is a shift of 1 in the grading (compared to the indexing of [[homotopy group]]s), so <math>\pi_k(X)</math> has degree <math>(k-1)</math>; equivalently, <math>L_k = \pi_{k+1}(X)</math> (setting ''L'' to be the graded quasi-Lie algebra). Thus <math>L_0 = \pi_1(X)</math> acts on each graded component.
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| ==Properties==
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| The Whitehead product is bilinear, graded-symmetric, and satisfies the [[graded Lie algebra|graded Jacobi identity]], and is thus a [[Graded Lie algebra|graded]] [[quasi-Lie algebra]]; this is proven in {{harvtxt|Uehara|Massey|1957}} via the [[Massey product|Massey triple product]].
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| <!-- (I don't know any example where <math>[f,f]\neq 0</math>.) -->
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| If <math>f \in \pi_1(X)</math>, then the Whitehead bracket is related to the usual conjugation action of <math>\pi_1</math> on <math>\pi_k</math> by
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| :<math>[f,g]=g^f-g, \, </math> | |
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| where <math>g^f</math> denotes the [[Inner automorphism|conjugation]] of <math>g</math> by <math>f</math>.
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| For <math>k=1</math>, this reduces to
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| :<math>[f,g]=fgf^{-1}g^{-1}, \,</math>
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| which is the usual commutator.
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| The relevant [[Mathematics Subject Classification|MSC]] code is: 55Q15, Whitehead products and generalizations.
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| ==See also==
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| * [[Generalised Whitehead product]]
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| * [[Massey product]]
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| * [[Toda bracket]]
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| ==References==
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| *{{citation|mr=0091473
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| |last=Uehara|first= Hiroshi|last2= Massey|first2= W. S.
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| |authorlink2=William Massey
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| |chapter=The Jacobi identity for Whitehead products|title= Algebraic geometry and topology. A symposium in honor of S. Lefschetz|pages=361–377|publisher= [[Princeton University Press]]|publication-place= Princeton, N. J.,|year= 1957}}
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| * {{citation
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| |first=George W.
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| |last=Whitehead
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| |authorlink=George W. Whitehead
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| |title=On products in homotopy groups
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| |journal=[[Annals of Mathematics]]
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| |series=2
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| |volume=47
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| |issue=3
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| |date=July 1946
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| |pages=460–475
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| |doi=10.2307/1969085
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| |jstor=1969085}}
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| * {{citation
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| |first=J. H. C.
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| |last=Whitehead
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| |authorlink=J. H. C. Whitehead
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| |title=On adding relations to homotopy groups
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| |journal=[[Annals of Mathematics]]
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| |series=2
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| |volume=42
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| |issue=2
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| |date=April 1941
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| |pages=409–428
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| |doi=10.2307/1968907
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| |jstor=1968907}}
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| [[Category:Homotopy theory]]
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| [[Category:Lie algebras]]
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Alyson Meagher is the title her mothers and fathers gave her but she doesn't like when people use her complete name. The favorite hobby for him and his children is to perform lacross and he would never give it up. Since he was 18 he's been operating as an info officer but he plans on altering it. Alaska is where I've always been residing.
My web blog - online psychic (you can try here)