nLab Whitehead product



Homotopy theory

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Basic definition


For XX a pointed topological space (CW-complex), the Whitehead products (Whitehead 41, Section 3) are the bilinear maps on the elements of the homotopy groups π (X)\pi_\bullet(X) of XX of the form

(1)[,] Wh:π n 1(X) π n 2(X)π n 1+n 21(X)AAfor alln i;n i1, [-,-]_{Wh} \;\colon\; \pi_{n_1}\big( X \big) \otimes_{\mathbb{Z}} \pi_{n_2}\big( X \big) \longrightarrow \pi_{n_1 + n_2 - 1}\big( X \big) \phantom{AA} \text{for all} \; n_i \in \mathbb{N} ; \; n_i \geq 1 \,,

given by sending any pair of homotopy classes

[S n iϕ iX]π n i(X) \big[ S^{n_i} \overset{\phi_i}{\longrightarrow} X \big] \;\in\; \pi_{ n_i } \big( X \big)

to the homotopy class of the top composite in the diagram

S n 1+n 21 f n 1,n 2 S n 1S n 2 (ϕ 1,ϕ 2) X (po) D n 1+n 2 S n 1×S n 2 \array{ S^{ n_1 + n_2 -1 } & \overset{ f_{n_1,n_2} }{ \longrightarrow } & S^{n_1} \vee S^{n_2} & \overset{ (\phi_1, \phi_2) }{\longrightarrow} & X \\ \big\downarrow & (po) & \big\downarrow \\ D^{ n_1 + n_2 } &\underset{}{\longrightarrow}& S^{n_1} \times S^{n_2} }

where f n 1,n 2f_{n_1, n_2} is the attaching map exhibiting the product space S n 1×S n 2S^{n_1} \times S^{n_2} as the result of a cell attachment to the wedge sum S n 1S n 2S^{n_1} \vee S^{n_2}.

In this form this appears for instance in Félix-Halperin-Thomas, p. 176 with p. 177.

Generalized version

There is also a generalized Whitehead product where we can take more general homotopy classes (continuous maps up to homotopy) α[S Y,X]\alpha\in [S^\cdot Y,X] and β[S Z,X]\beta\in [S^\cdot Z,X] to produce a class [α,β] Wh[YZ,X][\alpha,\beta]_{Wh}\in[Y\star Z,X]. Here S S^\cdot denotes the reduced suspension operation on pointed spaces and \star denotes the join of CW-complexes. Notice that ptZ=C (Z)pt\star Z = C^\cdot(Z) and the reduced cone of a point is C (pt)=S 1C^\cdot(pt)=S^1. Thus for Y=Z=ptY=Z=pt the generalized Whitehead product reduced to the usual Whitehead product.


Super Lie algebra structure

If one assigns degree n1n-1 to the nnth homotopy group π n\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:

(2)π +1(X)= π 1(X) π 2(X) π 3(X) deg= 0 1 2 \array{ \pi_{\bullet + 1}(X) = & \pi_1(X) &\oplus& \pi_2(X) &\oplus& \pi_3(X) &\oplus& \cdots \\ deg = & 0 && 1 && 2 }

This unified Whitehead product is graded skew-symmetric in that for ϕ iπ n i(X)\phi_i \in \pi_{n_i}\big( X \big) it satisfies

[ϕ 1,ϕ 2] Wh=(1) n 1n 2[ϕ 2,ϕ 1] Wh \big[ \phi_1, \, \phi_2 \big]_{Wh} \;=\; (-1)^{ n_1 n_2 } \big[ \phi_2, \, \phi_1 \big]_{Wh}

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 π 1(X)\pi_{\bullet-1}(X) (2), over the ring of integers (sometimes called, in this context, a graded quasi-Lie algebra, see below).

Of elements with themselves

Beware that the skew-symmetry of Lie algebras over the integers, as opposed to over a field of characteristic zero, implies for any element ϕ\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

[ϕ,ϕ] Wh=[ϕ,ϕ] WhAAhence equivalentlyAA2[ϕ,ϕ] Wh=0 [\phi,\phi]_{Wh} = - [\phi,\phi]_{Wh} \phantom{AA} \text{hence equivalently} \phantom{AA} 2 \cdot [\phi,\phi]_{Wh} = 0

but not necessarily that [ϕ,ϕ] Wh=0[\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.

As primary homotopy operations

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.

Relation to Pontrjagin product

Under the Hurewicz homomorphism, the Whitehead product on homotopy groups is the commutator of the Pontrjagin product on integral homology groups of a based loop space.

This is due to Samelson (1953) and for higher Whitehead brackets due to Arkowitz (1971).

In fact, in characteristic zero the Pontrjagin ring is the universal enveloping algebra of the Whitehead bracket Lie algebra [Milnor & Moore (1965) Appendix].

A textbook account is in Whitehead (1978) Thm. X.7.10.

In terms of Samelson products

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)

Relation to the Sullivan models

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 WW a \mathbb{Z}-graded module, we write

WWSym 2(W)(WW)/(αβ(1) n αn ββα), W \wedge W \;\coloneqq\; Sym^2(W) \;\coloneqq\; \big( W \otimes W \big) / \big( \alpha \otimes \beta \sim (-1)^{ n_\alpha n_\beta } \beta \otimes \alpha \big) \,,

where on the right α,βW\alpha, \beta \in W are elements of homogeneous degree n α,n βn_\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 XX be a simply connected topological space with Sullivan model

(3)CE(𝔩X)=(Sym (V *),d X) CE( \mathfrak{l} X ) \;=\; \big( Sym^\bullet\big(V^\ast\big), d_X \big)

for V *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 XX:

V *Hom Ab(π (X),). V^\ast \;\coloneqq\; Hom_{Ab}\big( \pi_\bullet(X), \mathbb{Q} \big) \,.

Declare the wedge product pairing to be given by

(4)V *V * Φ Hom Ab(π (X)π (X),) (α,β) (vw(1) n αn βα(v)β(w)+β(v)α(w)) \array{ V^\ast \wedge V^\ast &\overset{\Phi}{\longrightarrow}& Hom_{Ab} \big( \pi_\bullet(X) \wedge \pi_\bullet(X) , \mathbb{Q} \big) \\ (\alpha, \beta) &\mapsto& \Big( v \wedge w \;\mapsto\; (-1)^{ n_\alpha \cdot n_\beta } \alpha(v)\cdot \beta(w) + \beta(v)\cdot \alpha(w) \Big) }

where α\alpha, β\beta are assumed to be of homogeneous degree n α,n βn_\alpha, n_\beta \in \mathbb{N}, respectively.

(Notice that the usual normalization factor of 1/21/2 is not included on the right. This normalization follows Andrews & Arkowitz 1978, above Thm. 6.1.)

Finally, write

(5)[] 2:Sym (V *)V *V * [-]_2 \;\colon\; Sym^\bullet\big(V^\ast\big) \longrightarrow V^\ast \wedge V^\ast

for the linear projection on quadratic polynomials in the graded symmetric algebra.



(co-binary Sullivan differential is Whitehead product)

Let XX be a simply connected topological space of rational finite type, so that it has a Sullivan model with Sullivan differential d Xd_X (3).

Then the co-binary component (5) of the Sullivan differential equals the \mathbb{Q}-linear dual map of the Whitehead product [,] X[-,-]_X on the homotopy groups of XX:

[d Xα] 2=[,] X *. [d_X \alpha]_2 \;=\; [-,-]_X^\ast \,.

More explicitly, the following diagram commutes:

V * [] 2d X V *V * = Φ Hom Ab(π (X),) Hom Ab([,] X,) Hom Ab(π (X)π (X),), \array{ V^\ast &\overset{ [-]_2\circ d_X }{\longrightarrow}& V^\ast \wedge V^\ast \\ \big\downarrow\mathrlap{^=} && \big\downarrow\mathrlap{^\Phi} \\ Hom_{Ab} \big( \pi_\bullet(X), \mathbb{Q} \big) & \underset{ Hom_{Ab}\big( [-,-]_X , \; \mathbb{Q} \big) }{ \longrightarrow } & Hom_{Ab} \big( \pi_\bullet(X) \wedge \pi_\bullet(X), \; \mathbb{Q} \big) } \,,

where the wedge product on the right is normalized as in (4).

(Andrews & Arkowitz 1978, Thm. 6.1, following Deligne, Griffiths, Morgan & Sullivan 1975)


Prop. says in particular that the binary bracket of the L L_\infty -algebra dual to a Sullivan model is always an actual super Lie bracket in that it satisfies its super-Jacobi identity, even if there happens to also be a nontrivial trinary bracket which would serve as a “Jacobiator”.

This is due to the minimality of Sullivan models, which implies that the co-unary part of their differential vanishes, and hence that that the unary bracket of the corresponding L L_\infty-algebra vanishes: Since the failure of the Jacobi identity on binary brackets in an L L_\infty-algebra is measured not by the trinary bracket itself but by its composition with the unary bracket, this vanishes in the above case.

Relation to Goodwillie Calculus:

On the relation to Goodwillie calculus see e.g. Scherer & Chorny 2011, 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 2X = S^2 the 2-sphere, consider the following two elements of its homotopy groups (of spheres, as it were):

  1. id S 2π 2(S 2)id_{S^2} \in \pi_2\big( S^2 \big) (represented by the identity function S 2S 2S^2 \to S^2)

  2. h π 3(S 2)h_{\mathbb{C}} \in \pi_3\big( S^2 \big) (represented by the complex Hopf fibration)

Then the Whitehead product satisfies

[id S 2,id S 2]=2h . \big[ id_{S^2}, \; id_{S^2} \big] \;=\; 2 \cdot h_{\mathbb{C}} \,.



The concept is originally due to

  • J. H. C. Whitehead, Section 3 of On Adding Relations to Homotopy Groups, Annals of Mathematics Second Series, Vol. 42, No. 2 (Apr., 1941), pp. 409-428 (jstor:1968907)

with early discussion in

Proof that the Whitehead product is the commutator of the Pontrjagin product:

and in characteristic zero:

and for higher Whitehead brackets:

Textbook account:

See also

Discussion of Whitehead products specifically of homotopy groups of spheres:

As n-excisive functors:

Discussion of Whitehead products in homotopy type theory:

In rational homotopy theory

Discussion of Whitehead products in rational homotopy theory (the co-binary Sullivan differential is the dual Whitehead product):

The Whitehead product of P 1P 1P ×P \mathbb{C}P^1 \vee \mathbb{C}P^1 \to \mathbb{C}P^\infty \times \mathbb{C}P^\infty in relation to the Dolbeault complex:

  • Shamuel Auyeung, Jin-Cheng Guu, Jiahao Hu, pp. 4 of: On the algebra generated by μ¯\overline{\mu}, ¯\overline{\partial}, \partial, μ\mu, Complex Manifolds 10 1 (2023) [arXiv:2208.04890, doi:10.1515/coma-2022-0149]

Last revised on January 28, 2024 at 11:50:21. See the history of this page for a list of all contributions to it.