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 [math]\displaystyle{ f \in \pi_k(X), g \in \pi_l(X) }[/math], the Whitehead bracket
- [math]\displaystyle{ [f,g] \in \pi_{k+l-1}(X) }[/math]
is defined as follows:
The product [math]\displaystyle{ S^k \times S^l }[/math] can be obtained by attaching a [math]\displaystyle{ (k+l) }[/math]-cell to the wedge sum
- [math]\displaystyle{ S^k \vee S^l }[/math];
the attaching map is a map
- [math]\displaystyle{ S^{k+l-1} \stackrel{\phi}{\ \longrightarrow\ } S^k \vee S^l. }[/math]
Represent [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] by maps
- [math]\displaystyle{ f\colon S^k \to X }[/math]
and
- [math]\displaystyle{ g\colon S^l \to X, }[/math]
then compose their wedge with the attaching map, as
- [math]\displaystyle{ S^{k+l-1} \stackrel{\phi}{\ \longrightarrow\ } S^k \vee S^l \stackrel{f \vee g}{\ \longrightarrow\ } X . }[/math]
The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of
- [math]\displaystyle{ \pi_{k+l-1}(X). }[/math]
Grading
Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so [math]\displaystyle{ \pi_k(X) }[/math] has degree [math]\displaystyle{ (k-1) }[/math]; equivalently, [math]\displaystyle{ L_k = \pi_{k+1}(X) }[/math] (setting L to be the graded quasi-Lie algebra). Thus [math]\displaystyle{ L_0 = \pi_1(X) }[/math] acts on each graded component.
Properties
The Whitehead product satisfies the following properties:
- Bilinearity. [math]\displaystyle{ [f,g+h] = [f,g] + [f,h], [f+g,h] = [f,h] + [g,h] }[/math]
- Graded Symmetry. [math]\displaystyle{ [f,g]=(-1)^{pq}[g,f], f \in \pi_p X, g \in \pi_q X, p,q \geq 2 }[/math]
- Graded Jacobi identity. [math]\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 }[/math]
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) via the Massey triple product.
Relation to the action of [math]\displaystyle{ \pi_{1} }[/math]
If [math]\displaystyle{ f \in \pi_1(X) }[/math], then the Whitehead bracket is related to the usual action of [math]\displaystyle{ \pi_1 }[/math] on [math]\displaystyle{ \pi_k }[/math] by
- [math]\displaystyle{ [f,g]=g^f-g, }[/math]
where [math]\displaystyle{ g^f }[/math] denotes the conjugation of [math]\displaystyle{ g }[/math] by [math]\displaystyle{ f }[/math].
For [math]\displaystyle{ k=1 }[/math], this reduces to
- [math]\displaystyle{ [f,g]=fgf^{-1}g^{-1}, }[/math]
which is the usual commutator in [math]\displaystyle{ \pi_1(X) }[/math]. This can also be seen by observing that the [math]\displaystyle{ 2 }[/math]-cell of the torus [math]\displaystyle{ S^{1} \times S^{1} }[/math] is attached along the commutator in the [math]\displaystyle{ 1 }[/math]-skeleton [math]\displaystyle{ S^{1} \vee S^{1} }[/math].
Whitehead products on H-spaces
For a path connected H-space, all the Whitehead products on [math]\displaystyle{ \pi_{*}(X) }[/math] 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 [math]\displaystyle{ \alpha \in \pi_{i}(X) }[/math], [math]\displaystyle{ \beta \in \pi_{j}(X) }[/math] lie in the kernel of the suspension homomorphism [math]\displaystyle{ \Sigma \colon \pi_{i+j-1}(X) \to \pi_{i+j}(\Sigma X) }[/math]
Examples
- [math]\displaystyle{ [\mathrm{id}_{S^{2}} , \mathrm{id}_{S^{2}}] = 2 \cdot \eta \in \pi_3(S^{2}) }[/math], where [math]\displaystyle{ \eta \colon S^{3} \to S^{2} }[/math] is the Hopf map.
This can be shown by observing that the Hopf invariant defines an isomorphism [math]\displaystyle{ \pi_{3}(S^{2}) \cong \Z }[/math] and explicitly calculating the cohomology ring of the cofibre of a map representing [math]\displaystyle{ [\mathrm{id}_{S^{2}}, \mathrm{id}_{S^{2}}] }[/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.
See also
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
- 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
- Whitehead, George W. (July 1946), "On products in homotopy groups", Annals of Mathematics, 2 47 (3): 460–475, doi:10.2307/1969085
- 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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