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.
Contents
Definition
Given elements
f
∈
π
k
(
X
)
,
g
∈
π
l
(
X
)
{\displaystyle f\in \pi _{k}(X),g\in \pi _{l}(X)}
, the Whitehead bracket
[
f
,
g
]
∈
π
k
+
l
−
1
(
X
)
{\displaystyle [f,g]\in \pi _{k+l-1}(X)}
is defined as follows:
The product
S
k
×
S
l
{\displaystyle S^{k}\times S^{l}}
can be obtained by attaching a
(
k
+
l
)
{\displaystyle (k+l)}
-cell to the wedge sum
S
k
∨
S
l
{\displaystyle S^{k}\vee S^{l}}
;
the attaching map is a map
S
k
+
l
−
1
⟶
ϕ
S
k
∨
S
l
.
{\displaystyle S^{k+l-1}{\stackrel {\phi }{\ \longrightarrow \ }}S^{k}\vee S^{l}.}
Represent
f
{\displaystyle f}
and
g
{\displaystyle g}
by maps
f
:
S
k
→
X
{\displaystyle f\colon S^{k}\to X}
and
g
:
S
l
→
X
,
{\displaystyle g\colon S^{l}\to X,}
then compose their wedge with the attaching map, as
S
k
+
l
−
1
⟶
ϕ
S
k
∨
S
l
⟶
f
∨
g
X
.
{\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
π
k
+
l
−
1
(
X
)
.
{\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
π
k
(
X
)
{\displaystyle \pi _{k}(X)}
has degree
(
k
−
1
)
{\displaystyle (k-1)}
; equivalently,
L
k
=
π
k
+
1
(
X
)
{\displaystyle L_{k}=\pi _{k+1}(X)}
(setting L to be the graded quasi-Lie algebra). Thus
L
0
=
π
1
(
Properties
The Whitehead product satisfies the following properties:
Bilinearity.
[
f
,
g
+
h
]
=
[
f
,
g
]
+
[
f
,
h
]
,
[
f
+
g
,
h
]
=
[
f
,
h
]
+
[
Relation to the action of
π
1
{\displaystyle \pi _{1}}
If
f
∈
π
1
(
X
)
{\displaystyle f\in \pi _{1}(X)}
, then the Whitehead bracket is related to the usual action of
π
1
{\displaystyle \pi _{1}}
on
π
k
{\displaystyle \pi _{k}}
by
[
f
,
g
]
=
g
f
−
g
,
{\displaystyle [f,g]=g^{f}-g,}
where
Whitehead products on H-spaces
For a path connected H-space, all the Whitehead products on
π
∗
(
X
)
{\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
α
∈
π
i
(
X
)
{\displaystyle \alpha \in \pi _{i}(X)}
,
β
∈
π
j
(
X
)
{\displaystyle \beta \in \pi _{j}(X)}
lie in the kernel of the suspension homomorphism
Σ
:
π
i
+
j
−
1
(
X
)
→
π
i
+
j
(
Examples
[
i
d
S
2
,
i
d
S
2
]
=
2
⋅
η
∈
π
3
(
S
2
)
{\displaystyle [\mathrm {id} _{S^{2}},\mathrm {id} _{S^{2}}]=2\cdot \eta \in \pi _{3}(S^{2})}
, where
η
:
S
3
→
S
2
{\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