nLab Whitehead product

Idea

The Whitehead bracket on the homotopy groups of a connected topological space is a reflection of the “group commutator” (Samelson product) on its grouplike loop space (cf. below).

Definition

Basic definition

Definition

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

(1)$$[-,-]_{Wh} \;\colon\; \pi_{n_1}\big( X \big) \otimes_{\mathbb{Z}} \pi_{n_2}\big( X \big) \longrightarrow \pi_{n_1 + n_2 - 1}\big( X \big) \phantom{AA} \text{for all} \; n_i \in \mathbb{N} ; \; n_i \geq 1 \,,$$

given by sending any pair of homotopy classes

$$\big[ S^{n_i} \overset{\phi_i}{\longrightarrow} X \big] \;\in\; \pi_{ n_i } \big( X \big)$$

to the homotopy class of the top composite in this diagram:

(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 $f_{n_1, n_2}$ 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 (2000) p. 176 with p. 177.

Example

For $n_1 = n_2 = 1$, the product space $S^1 \times S^1$ is the torus and the attaching map $f_{1,1}$ in (2) is a representative of the group commutator of a pair of generators of the fundamental group of its 1-skeleton (see at surface, this paragraph).

Indeed, the Whitehead product is generally given by forming group commutators: see below at In terms of Samelson products.

Generalized version

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.

Properties

General

Proposition

The suspension of any Whitehead product is null.

(This is Chapter X, Theorem 8.20 (p. 485) in Whitehead 1978, using the notation “$E$” for the suspension functor which is introduced on p. 369, right before Theorem 7.13 there. The statement is reiterated in words on p .549, above Cor. 2.6 there.)

Super Lie algebra structure

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:

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

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

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

(Whitehead 1978 (7.5) on p. 474) and it satisfies the corresponding graded Jacobi identity (Hilton 1955 Thm. B. Whitehead 1978 (7.14) on p. 478).

This makes the Whitehead bracket the Lie bracket of a super Lie algebra structure on $\pi_{\bullet-1}(X)$ (3), over the ring of integers (sometimes called, in this context, a graded quasi-Lie algebra, see below).

Remark

(Whithead bracket of elements with themselves)

It follows in particular that the Whitehead bracket of even-degree homotopy groups with themselves need not vanish by degree reasons — notably $\big[ [id_{S^{2k}}], [id_{S^{2k}}]\big]$ is non-vanishing (and of Hopf invariant 2, cf. Prop. ).

On the other hand, the skew-symmetry of Lie algebras over the integers, as opposed to over a field of characteristic zero, also 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

$$[\phi,\phi]_{Wh} = - [\phi,\phi]_{Wh} \phantom{AA} \text{hence equivalently} \phantom{AA} 2 \cdot [\phi,\phi]_{Wh} = 0 \mathrlap{\,,}$$

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.

(cf. Whitehead 1978, Thm. 8.8 on p. 536)

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

For $X$ a connected pointed topological space, consider its loop space

$$\mathcal{G} \coloneqq \Omega X \mathrlap{\,,}$$

regarded as a grouplike space with a Samelson product $\langle-,-\rangle$ (essentially the group commutator of $\mathcal{G}$), whence (by the looping and delooping relation)

$$X \simeq B \mathcal{G} \mathrlap{\,.}$$

Then we have:

1. the Samelson product on the homotopy groups of $\mathcal{G}$:

   $$\langle -, - \rangle \;\colon\; \pi_n(\mathcal{G}) \times \pi_m(\mathcal{G}) \longrightarrow \pi_{n+m}(\mathcal{G}) \mathrlap{\,.}$$

2. the Whitehead bracket on the homotopy groups of $X$:

   $$[- , - ] \;\colon\; \pi_n(X) \times \pi_m(X) \longrightarrow \pi_{n+m-1}(X) \mathrlap{\,,}$$

3. the canonical isomorphism

   (4)$$\tau \,\colon\, \pi_{n+1}(X) \xrightarrow{ \sim } \pi_n(\mathcal{G}) \mathrlap{\,.}$$

Proposition

Under the isomorphism (4) the Samelson product on the homotopy groups of $\mathcal{G}$ coincides up to a sign with the Whitehead bracket on the homotopy groups of $X \simeq B \mathcal{G}$, in that for pairs $\alpha_p \in \pi_{p+1}(X)$, $\beta_q \in \pi_{q+1}(X)$ we have

$$\tau[\alpha_p, \beta_q] \;=\; (-1)^p \big\langle \tau \alpha_p ,\, \tau \beta_q \big\rangle \mathrlap{\,.}$$

(cf. Whitehead 1978 §X.7 Thm 7.10 (p. 476))

Remark

In the context of simplicial homotopy theory, with 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 that is a kind 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 $W$ a $\mathbb{Z}$-graded module, we write

$$W \wedge W \;\coloneqq\; Sym^2(W) \;\coloneqq\; \big( W \otimes W \big) / \big( \alpha \otimes \beta \sim (-1)^{ n_\alpha n_\beta } \beta \otimes \alpha \big) \,,$$

where on the right $\alpha, \beta \in W$ 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

(5)$$CE( \mathfrak{l} X ) \;=\; \big( Sym^\bullet\big(V^\ast\big), d_X \big)$$

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 $X$:

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

Declare the wedge product pairing to be given by

(6)$$\array{ V^\ast \wedge V^\ast &\overset{\Phi}{\longrightarrow}& Hom_{Ab} \big( \pi_\bullet(X) \wedge \pi_\bullet(X) , \mathbb{Q} \big) \\ (\alpha, \beta) &\mapsto& \Big( v \wedge w \;\mapsto\; (-1)^{ n_\alpha \cdot n_\beta } \alpha(v)\cdot \beta(w) + \beta(v)\cdot \alpha(w) \Big) }$$

where $\alpha$, $\beta$ 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 1978, above Thm. 6.1.)

Finally, write

(7)$$[-]_2 \;\colon\; Sym^\bullet\big(V^\ast\big) \longrightarrow V^\ast \wedge V^\ast$$

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

Then:

Proposition

(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 $d_X$ (5).

Then the co-binary component (7) of the Sullivan differential equals the $\mathbb{Q}$-linear dual map of the Whitehead product $[-,-]_X$ on the homotopy groups of $X$:

$$[d_X \alpha]_2 \;=\; [-,-]_X^\ast \,.$$

More explicitly, the following diagram commutes:

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

where the wedge product on the right is normalized as in (6).

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

Remark

Prop. says in particular that the binary bracket of the $L_\infty$-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 $L_\infty$-algebra vanishes: Since the failure of the Jacobi identity on binary brackets in an $L_\infty$-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.

Examples

Example

(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):

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

2. $[h_{\mathbb{C}}] \in \pi_3\big( S^2 \big)$ (represented by the complex Hopf fibration)

Then the Whitehead product satisfies

$$\big[ [id_{S^2}], \; [id_{S^2}] \big] \;=\; 2 \cdot [h_{\mathbb{C}}] \eqqcolon 2 \eta \,.$$

Generally:

Proposition

For $k \in \mathbb{N}$,

$$\big[ [id_{S^{2k}}], \; [id_{S^{2k}}] \big]$$

has Hopf invariant 2

(Whitehead 1978 Thm. 2.5 on p. 495, using the notation $\iota_n \coloneqq [id_{S^{n}}]$ introduced on p. 194).

So we have similarly that

$$\big[ [id_{S^4}], \; [id_{S^4}] \big] \;=\; 2 \cdot [h_{\mathbb{H}}] \eqqcolon 2 \nu$$

for $h_{\mathbb{H}}$ the quaternionic Hopf fibration, and

$$\big[ [id_{S^8}], \; [id_{S^8}] \big] \;=\; 2 \cdot [h_{\mathbb{O}}] \eqqcolon 2 \sigma$$

for $h_{\mathbb{O}}$ the octonionic Hopf fibration.

Related concepts

• Pontrjagin product

• Samelson product

References

General

The concept is due to

• J. H. C. Whitehead, Section 3 ofL On Adding Relations to Homotopy Groups, Annals of Mathematics Second Series, 42 2 (1941) 409-428 [jstor:1968907]

with further early discussion in:

• George W. Whitehead: On Products in Homotopy Groups, Annals of Mathematics 47 3 (1946) 460-475 [doi:10.2307/1969085, jstor:1969085]

• Peter Hilton, J. H. C. Whitehead: Note on the Whitehead Product, Annals of Mathematics Second Series 58 3 (1953) 429-442 [jstor:1969746]

• Peter Hilton, On the homotopy groups of unions of spheres, J. London Math. Soc., 30 (1955) 154–172 [pdf, doi:doi.org/10.1112/jlms/s1-30.2.154]

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

• Hans Samelson, A Connection Between the Whitehead and the Pontryagin Product, American Journal of Mathematics 75 4 (1953) 744–752 [doi:10.2307/2372549]

and in characteristic zero:

• John Milnor, John Moore, Appendix of: On the structure of Hopf algebras, Annals of Math. 81 (1965) 211-264 [doi:10.2307/1970615, pdf]

and for higher Whitehead brackets:

• Martin Arkowitz, Whitehead Products as Images of Pontrjagin Products, Transactions of the American Mathematical Society, 158 2 (1971) 453-463 [doi:10.2307/1995917]

Textbook account:

• George W. Whitehead, §X.7 in: Elements of Homotopy Theory, Springer (1978) [doi:10.1007/978-1-4612-6318-0]

See also

• Wikipedia Whitehead product

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)

As n-excisive functors:

• Jerome Scherer, Boris Chorny, Goodwillie calculus and Whitehead products, Forum Math. 27 (2015), no. 1, 119 - 130 (arXiv:1109.2691)

Discussion of Whitehead products in homotopy type theory:

• Guillaume Brunerie, section 3.3 of On the homotopy groups of spheres in homotopy type theory (arXiv:1606.05916)

In rational homotopy 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 (1978) 961-982 [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 (2000) [doi:10.1007/978-1-4613-0105-9]

• 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)

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

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