In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.
In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map
η
:
S
3
→
S
2
,
and proved that
η
is essential, i.e. not homotopic to the constant map, by using the linking number (=1) of the circles
η
−
1
(
x
)
,
η
−
1
(
y
)
⊂
S
3
for any
x
≠
y
∈
S
2
.
It was later shown that the homotopy group
π
3
(
S
2
)
is the infinite cyclic group generated by
η
. In 1951, Jean-Pierre Serre proved that the rational homotopy groups
π
i
(
S
n
)
⊗
Q
for an odd-dimensional sphere (
n
odd) are zero unless i = 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree
2
n
−
1
.
Let
ϕ
:
S
2
n
−
1
→
S
n
be a continuous map (assume
n
>
1
). Then we can form the cell complex
C
ϕ
=
S
n
∪
ϕ
D
2
n
,
where
D
2
n
is a
2
n
-dimensional disc attached to
S
n
via
ϕ
. The cellular chain groups
C
c
e
l
l
∗
(
C
ϕ
)
are just freely generated on the
i
-cells in degree
i
, so they are
Z
in degree 0,
n
and
2
n
and zero everywhere else. Cellular (co-)homology is the (co-)homology of this chain complex, and since all boundary homomorphisms must be zero (recall that
n
>
1
), the cohomology is
H
c
e
l
l
i
(
C
ϕ
)
=
{
Z
i
=
0
,
n
,
2
n
,
0
otherwise
.
Denote the generators of the cohomology groups by
H
n
(
C
ϕ
)
=
⟨
α
⟩
and
H
2
n
(
C
ϕ
)
=
⟨
β
⟩
.
For dimensional reasons, all cup-products between those classes must be trivial apart from
α
⌣
α
. Thus, as a ring, the cohomology is
H
∗
(
C
ϕ
)
=
Z
[
α
,
β
]
/
⟨
β
⌣
β
=
α
⌣
β
=
0
,
α
⌣
α
=
h
(
ϕ
)
β
⟩
.
The integer
h
(
ϕ
)
is the Hopf invariant of the map
ϕ
.
Theorem:
h
:
π
2
n
−
1
(
S
n
)
→
Z
is a homomorphism. Moreover, if
n
is even,
h
maps onto
2
Z
.
The Hopf invariant is
1
for the Hopf maps (where
n
=
1
,
2
,
4
,
8
, corresponding to the real division algebras
A
=
R
,
C
,
H
,
O
, respectively, and to the fibration
S
(
A
2
)
→
P
A
1
sending a direction on the sphere to the subspace it spans). It is a theorem, proved first by Frank Adams and subsequently by Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1.
A very general notion of the Hopf invariant can be defined, but it requires a certain amount of homotopy theoretic groundwork:
Let
V
denote a vector space and
V
∞
its one-point compactification, i.e.
V
≅
R
k
and
V
∞
≅
S
k
for some
k
.
If
(
X
,
x
0
)
is any pointed space (as it is implicitly in the previous section), and if we take the point at infinity to be the basepoint of
V
∞
, then we can form the wedge products
V
∞
∧
X
.
Now let
F
:
V
∞
∧
X
→
V
∞
∧
Y
be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of
F
is
h
(
F
)
∈
{
X
,
Y
∧
Y
}
Z
2
,
an element of the stable
Z
2
-equivariant homotopy group of maps from
X
to
Y
∧
Y
. Here "stable" means "stable under suspension", i.e. the direct limit over
V
(or
k
, if you will) of the ordinary, equivariant homotopy groups; and the
Z
2
-action is the trivial action on
X
and the flipping of the two factors on
Y
∧
Y
. If we let
Δ
X
:
X
→
X
∧
X
denote the canonical diagonal map and
I
the identity, then the Hopf invariant is defined by the following:
h
(
F
)
:=
(
F
∧
F
)
(
I
∧
Δ
X
)
−
(
I
∧
Δ
Y
)
(
I
∧
F
)
.
This map is initially a map from
V
∞
∧
V
∞
∧
X
to
V
∞
∧
V
∞
∧
Y
∧
Y
,
but under the direct limit it becomes the advertised element of the stable homotopy
Z
2
-equivariant group of maps. There exists also an unstable version of the Hopf invariant
h
V
(
F
)
, for which one must keep track of the vector space
V
.
