Whitehead product

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 $f\in \pi _{k}(X),g\in \pi _{l}(X)$ , the Whitehead bracket

[f,g]\in \pi _{k+l-1}(X)

is defined as follows:

The product $S^{k}\times S^{l}$ can be obtained by attaching a $(k+l)$ -cell to the wedge sum

S^{k}\vee S^{l}

;

the attaching map is a map

S^{k+l-1}{\stackrel {\phi }{\ \longrightarrow \ }}S^{k}\vee S^{l}.

Represent $f$ and $g$ by maps

f\colon S^{k}\to X

and

g\colon S^{l}\to X,

then compose their wedge with the attaching map, as

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

\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 $\pi _{k}(X)$ has degree $(k-1)$ ; equivalently, $L_{k}=\pi _{k+1}(X)$ (setting L to be the graded quasi-Lie algebra). Thus $L_{0}=\pi _{1}(X)$ acts on each graded component.

Properties

The Whitehead product satisfies the following properties:

Bilinearity. $[f,g+h]=[f,g]+[f,h],[f+g,h]=[f,h]+[g,h]$
Graded Symmetry. $[f,g]=(-1)^{pq}[g,f],f\in \pi _{p}X,g\in \pi _{q}X,p,q\geq 2$
Graded Jacobi identity. $(-1)^{pr}[[f,g],h]+(-1)^{pq}[[g,h],f]+(-1)^{rq}[[h,f],g]=0,f\in \pi _{p}X,g\in \pi _{q}X,h\in \pi _{r}X{\text{ with }}p,q,r\geq 2$

Sometimes the homotopy groups of a space, together with the Whitehead product operation are called a graded quasi-Lie algebra; this is proven in Uehara & Massey (1957) via the Massey triple product.

Relation to the action of $\pi _{1}$

If $f\in \pi _{1}(X)$ , then the Whitehead bracket is related to the usual action of $\pi _{1}$ on $\pi _{k}$ by

[f,g]=g^{f}-g,

where $g^{f}$ denotes the conjugation of $g$ by $f$ .

For $k=1$ , this reduces to

[f,g]=fgf^{-1}g^{-1},

which is the usual commutator in $\pi _{1}(X)$ . This can also be seen by observing that the $2$ -cell of the torus $S^{1}\times S^{1}$ is attached along the commutator in the $1$ -skeleton $S^{1}\vee S^{1}$ .

Whitehead products on H-spaces

For a path connected H-space, all the Whitehead products on $\pi _{*}(X)$ vanish. By the previous subsection, this is a generalization of both the facts that the fundamental groups of H-spaces are abelian, and that H-spaces are simple.

Suspension

All Whitehead products of classes $\alpha \in \pi _{i}(X)$ , $\beta \in \pi _{j}(X)$ lie in the kernel of the suspension homomorphism $\Sigma \colon \pi _{i+j-1}(X)\to \pi _{i+j}(\Sigma X)$

Examples

$[\mathrm {id} _{S^{2}},\mathrm {id} _{S^{2}}]=2\cdot \eta \in \pi _{3}(S^{2})$ , where $\eta \colon S^{3}\to S^{2}$ is the Hopf map.

This can be shown by observing that the Hopf invariant defines an isomorphism $\pi _{3}(S^{2})\cong \mathbb {Z}$ and explicitly calculating the cohomology ring of the cofibre of a map representing $[\mathrm {id} _{S^{2}},\mathrm {id} _{S^{2}}]$ . Using the Pontryagin–Thom construction there is a direct geometric argument, using the fact that the preimage of a regular point is a copy of the Hopf link.

Applications to ∞-groupoids

Recall that an ∞-groupoid $\Pi (X)$ is an $\infty$ -category generalization of groupoids which is conjectured to encode the data of the homotopy type of $X$ in an algebraic formalism. The objects are the points in the space $X$ , morphisms are homotopy classes of paths between points, and higher morphisms are higher homotopies of those points.

The existence of the Whitehead product is the main reason why defining a notion of ∞-groupoids is such a demanding task. It was shown that any strict ∞-groupoid^[1] has only trivial Whitehead products, hence strict groupoids can never model the homotopy types of spheres, such as $S^{3}$ .^[2]

References

↑ Brown, Ronald; Higgins, Philip J. (1981). "The equivalence of ∞-groupoids and crossed complexes". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 22 (4): 371–386.
↑ Simpson, Carlos (1998-10-09). "Homotopy types of strict 3-groupoids". arXiv:math/9810059.

Whitehead, J. H. C. (April 1941), "On adding relations to homotopy groups", Annals of Mathematics, 2, 42 (2): 409–428, doi:10.2307/1968907, JSTOR 1968907
Uehara, Hiroshi; Massey, William S. (1957), "The Jacobi identity for Whitehead products", Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton, N. J.: Princeton University Press, pp. 361–377, MR 0091473
Whitehead, George W. (July 1946), "On products in homotopy groups", Annals of Mathematics, 2, 47 (3): 460–475, doi:10.2307/1969085, JSTOR 1969085
Whitehead, George W. (1978). "X.7 The Whitehead Product". Elements of homotopy theory. Springer-Verlag. pp. 472–487. ISBN 978-0387903361.

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[1] Brown, Ronald; Higgins, Philip J. (1981). "The equivalence of ∞-groupoids and crossed complexes". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 22 (4): 371–386.

[2] Simpson, Carlos (1998-10-09). "Homotopy types of strict 3-groupoids". arXiv:math/9810059.

[1]

[2]