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

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

is defined as follows:

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

${\displaystyle S^{k}\vee S^{l}}$;

the attaching map is a map

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

Represent ${\displaystyle f}$ and ${\displaystyle g}$ by maps

${\displaystyle f\colon S^{k}\to X}$

and

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

then compose their wedge with the attaching map, as

${\displaystyle S^{k+l-1}{\stackrel {\phi }{\ \longrightarrow \ }}S^{k}\vee S^{l}{\stackrel {f\vee g}{\ \longrightarrow \ }}X.}$

The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of

${\displaystyle \pi _{k+l-1}(X).}$

Grading[]

Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so ${\displaystyle \pi _{k}(X)}$ has degree ${\displaystyle (k-1)}$; equivalently, ${\displaystyle L_{k}=\pi _{k+1}(X)}$ (setting L to be the graded quasi-Lie algebra). Thus ${\displaystyle L_{0}=\pi _{1}(X)}$ acts on each graded component.

Properties[]

The Whitehead product satisfies the following properties:

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

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 ${\displaystyle \pi _{1}}$[]

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

${\displaystyle [f,g]=g^{f}-g,}$

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

For ${\displaystyle k=1}$, this reduces to

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

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

Whitehead products on H-spaces[]

For a path connected H-space, all the Whitehead products on ${\displaystyle \pi _{*}(X)}$ 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 ${\displaystyle \alpha \in \pi _{i}(X)}$, ${\displaystyle \beta \in \pi _{j}(X)}$ lie in the kernel of the suspension homomorphism ${\displaystyle \Sigma \colon \pi _{i+j-1}(X)\to \pi _{i+j}(\Sigma X)}$

Examples[]

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

This can be shown by observing that the Hopf invariant defines an isomorphism ${\displaystyle \pi _{3}(S^{2})\cong \mathbb {Z} }$ and explicitly calculating the cohomology ring of the cofibre of a map representing ${\displaystyle [\mathrm {id} _{S^{2}},\mathrm {id} _{S^{2}}]}$. 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 ${\displaystyle \Pi (X)}$ is an ${\displaystyle \infty }$-category generalization of groupoids which is conjectured to encode the data of the homotopy type of ${\displaystyle X}$ in an algebraic formalism. The objects are the points in the space ${\displaystyle X}$, 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 ${\displaystyle S^{3}}$.^[2]

See also[]

• Generalised Whitehead product
• Massey product
• Toda bracket

References[]

• 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.