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 .

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

: ,g\in \pi_(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

:$S^ \stackrel S^k \vee S^l.$

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

:$S^ \stackrel S^k \vee S^l \stackrel X .$

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

:$\pi_(X).$

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, $L_k = \pi_(X)$ (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. ,g+h= ,g+ ,h +g,h= ,h+ ,h/math>
* Graded Symmetry. ,g(-1)^ ,f f \in \pi_p X, g \in \pi_q X, p,q \geq 2
* Graded Jacobi identity. (-1)^ f,gh] + (-1)^ g,hf] + (-1)^ h,fg] = 0, f \in \pi_p X, g \in \pi_q X, h \in \pi_r X \text 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 via the Massey product, Massey triple product.

Relation to the action of $\pi_$

If $f \in \pi_1(X)$, then the Whitehead bracket is related to the usual action of $\pi_1$ on $\pi_k$ by

: ,gg^f-g,

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

For $k=1$, this reduces to

: ,gfgf^g^,

which is the usual commutator in $\pi_1(X)$. This can also be seen by observing that the $2$-cell of the torus $S^ \times S^$ is attached along the commutator in the $1$-skeleton $S^ \vee S^$.

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 $\alpha \in \pi_(X)$, $\beta \in \pi_(X)$ lie in the kernel of the suspension homomorphism $\Sigma \colon \pi_(X) \to \pi_(\Sigma X)$

Examples

* mathrm_ , \mathrm_= 2 \cdot \eta \in \pi_3(S^), where $\eta \colon S^ \to S^$ is the Hopf map.

This can be shown by observing that the Hopf invariant defines an isomorphism $\pi_(S^) \cong \Z$ and explicitly calculating the cohomology ring of the cofibre of a map representing mathrm_, \mathrm_/math>. 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 has only trivial Whitehead products, hence strict groupoids can never model the homotopy types of spheres, such as $S^3$.

See also

* Generalised Whitehead product
* Massey product
* Toda bracket

References

*
*
*
* {{cite book
, first=George W.
, last=Whitehead
, authorlink=George W. Whitehead
, title=Elements of homotopy theory
, chapter=X.7 The Whitehead Product
, publisher= Springer-Verlag
, isbn=978-0387903361
, pages=472–487, year=1978

Homotopy theory
Lie algebras