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For $X$ a pointed topological space (CW-complex), the Whitehead products (Whitehead 41, Section 3) are the bilinear maps on the elements of the homotopy groups $\pi_\bullet(X)$ of $X$ of the form
given by sending any pair of homotopy classes
to the homotopy class of the top composite in the diagram
where $f_{n_1, n_2}$ is the attaching map exhibiting the product space $S^{n_1} \times S^{n_2}$ as the result of a cell attachment to the wedge sum $S^{n_1} \vee S^{n_2}$.
In this form this appears for instance in Félix-Halperin-Thomas, p. 176 with p. 177.
There is also a generalized Whitehead product where we can take more general homotopy classes (continuous maps up to homotopy) $\alpha\in [S^\cdot Y,X]$ and $\beta\in [S^\cdot Z,X]$ to produce a class $[\alpha,\beta]_{Wh}\in[Y\star Z,X]$. Here $S^\cdot$ denotes the reduced suspension operation on pointed spaces and $\star$ denotes the join of CW-complexes. Notice that $pt\star Z = C^\cdot(Z)$ and the reduced cone of a point is $C^\cdot(pt)=S^1$. Thus for $Y=Z=pt$ the generalized Whitehead product reduced to the usual Whitehead product.
If one assigns degree $n-1$ to the $n$th homotopy group $\pi_n$, then the degree-wise Whitehead products (1) organize into a single degree-0 bilinear pairing on the graded abelian group which is the direct sum of all the homotopy groups:
This unified Whitehead product is graded skew-symmetric in that for $\phi_i \in \pi_{n_i}\big( X \big)$ it satisfies
and it satisfies the corresponding graded Jacobi identity (Hilton 55, Theorem B).
This makes the Whitehead bracket the Lie bracket of a super Lie algebra structure on $\pi_{\bullet-1}(X)$ (2), over the ring of integers (sometimes called, in this context, a graded quasi-Lie algebra, see below).
Beware that the skew-symmetry of Lie algebras over the integers, as opposed to over a field of characteristic zero, implies for any element $\phi$ of even homogeneous degree – hence here for elements of homotopy groups in odd degree – only that the bracket with itself vanishes after multiplication by 2
but not necessarily that $[\phi,\phi]_{Wh} = 0$ by itself – since multiplication by 2 is not an isomorphism over the integers.
But this means that the Whitehead bracket of any even-degree element with itself – hence of any element of a homotopy group in odd degree – has order at most 2, hence is in the 2-torsion subgroup of the respective homotopy group.
The Whitehead products form one of the primary homotopy operations.
In fact, together with composition operations and fundamental group-actions they generate all such operations.
This is related to the definition of Pi-algebras.
In the context of simplicial groups, representing connected homotopy types, there is a formula for the Whitehead product in terms of a Samelson product, which in turn is derived from a shuffle product which is a sort of non-commutative version of the Eilenberg-Zilber map. These simplicial formulae come from an analysis of the structure of the product of simplices.
(This formula for the Whitehead product is due to Dan Kan and can be found in the old survey article of Ed Curtis. The proof that it works was never published. For more pointers see MO:q/296479/381)
We discuss (Prop. below) how the rationalization of the Whitehead product is the co-binary part of the Sullivan differential in rational homotopy theory. First we make explicit some notation and normalization conventions that enter this statement:
In the following, for $W$ a $\mathbb{Z}$-graded module, we write
where on the right $\alpha, \beta \in W$ are elements of homogeneous degree $n_\alpha, n_\beta \in \mathbb{Z}$, respectively. The point is just to highlight that “$(-)\wedge(-)$” is not to imply here a degree shift of the generators (as it typically does in the usual notation for Grassmann algebras).
Let $X$ be a simply connected topological space with Sullivan model
for $V^\ast$ the graded vector space of generators, which is the $\mathbb{Q}$-linear dual graded vector space of the graded $\mathbb{Z}$-module (=graded abelian group) of homotopy groups of $X$:
Declare the wedge product pairing to be given by
where $\alpha$, $\beta$ are assumed to be of homogeneous degree $n_\alpha, n_\beta \in \mathbb{N}$, respectively.
(Notice that the usual normalization factor of $1/2$ is not included on the right. This normalization follows Andrews-Arkowitz 78, above Thm. 6.1.)
Finally, write
for the linear projection on quadratic polynomials in the graded symmetric algebra.
Then:
(co-binary Sullivan differential is Whitehead product)
Let $X$ be a simply connected topological space of rational finite type, so that it has a Sullivan model with Sullivan differential $d_X$ (3).
Then the co-binary component (5) of the Sullivan differential equals the $\mathbb{Q}$-linear dual map of the Whitehead product $[-,-]_X$ on the homotopy groups of $X$:
More explicitly, the following diagram commutes:
where the wedge product on the right is normalized as in (4).
(Andrews-Arkowitz 78, Thm. 6.1, following Deligne-Griffiths-Morgan-Sullivan 75)
See e.g. Scherer-Chorny 11, Sec. 1, which also gives an application of the relationship between the Whitehead and Samelson products.
(Whitehead product corresponding to complex Hopf fibration)
For $X = S^2$ the 2-sphere, consider the following two elements of its homotopy groups (of spheres, as it were):
$id_{S^2} \in \pi_2\big( S^2 \big)$ (represented by the identity function $S^2 \to S^2$)
$h_{\mathbb{C}} \in \pi_3\big( S^2 \big)$ (represented by the complex Hopf fibration)
Then the Whitehead product satisfies
The concept is originally due to
with early discussion in
Peter Hilton, J. H. C. Whitehead, Note on the Whitehead Product, Annals of Mathematics Second Series, Vol. 58, No. 3 (Nov., 1953), pp. 429-442 (jstor:1969746)
Peter Hilton, On the homotopy groups of unions of spheres, J. London Math. Soc., 1955, 30, 154–172 (pdf, doi:doi.org/10.1112/jlms/s1-30.2.154)
See also
Discussion of Whitehead products specifically of homotopy groups of spheres:
Ioan Mackenzie James, On the Suspension Triad, Annals of Mathematics Second Series, Vol. 63, No. 2 (Mar., 1956), pp. 191-247 (arXiv:1969607)
Ioan Mackenzie James, On the Suspension Sequence, Annals of Mathematics Second Series, Vol. 65, No. 1 (Jan., 1957), pp. 74-107 (arXiv:1969666)
Discussion of Whitehead products in homotopy type theory:
Discussion of Whitehead products in rational homotopy theory (the co-binary Sullivan differential is the dual Whitehead product):
Daniel Quillen, section I.5 of Rational Homotopy Theory, Annals of Mathematics Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (jstor:1970725)
Christopher Allday, Rational Whitehead products and a spectral sequence of Quillen, Pacific J. Math. Volume 46, Number 2 (1973), 313-323 (euclid:1102946308)
Christopher Allday, Rational Whitehead product and a spectral sequence of Quillen, II, Houston Journal of Mathematics, Volume 3, No. 3, 1977 (pdf)
Pierre Deligne, Phillip Griffiths, John Morgan, Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent Math (1975) 29: 245 (doi:10.1007/BF01389853)
Peter Andrews, Martin Arkowitz, Sullivan’s Minimal Models and Higher Order Whitehead Products, Canadian Journal of Mathematics, 30(5), 961-982, 1978 (doi:10.4153/CJM-1978-083-6)
Yves Félix, Steve Halperin, J.C. Thomas, Prop. 13.16 in Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.
Francisco Belchí, Urtzi Buijs, José M. Moreno-Fernández, Aniceto Murillo, Higher order Whitehead products and $L_\infty$ structures on the homology of a DGL, Linear Algebra and its Applications, Volume 520 (2017), pages 16-31 (arXiv:1604.01478, doi:10.1016/j.laa.2017.01.008)
Takahito Naito, A model for the Whitehead product in rational mapping spaces (arXiv:1106.4080)