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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
( R e d i r e c t e d f r o m H o p f i n v a r i a n t o n e )
Motivation [ edit ]
In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map
η
:
S
3
→
S
2
,
{\displaystyle \eta \colon S^{3}\to S^{2},}
and proved that
η
{\displaystyle \eta }
is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles
η
−
1
(
x
)
,
η
−
1
(
y
)
⊂
S
3
{\displaystyle \eta ^{-1}(x ),\eta ^{-1}(y )\subset S^{3}}
is equal to 1, for any
x
≠
y
∈
S
2
{\displaystyle x\neq y\in S^{2}}
.
It was later shown that the homotopy group
π
3
(
S
2
)
{\displaystyle \pi _{3}(S^{2})}
is the infinite cyclic group generated by
η
{\displaystyle \eta }
. In 1951, Jean-Pierre Serre proved that the rational homotopy groups [1]
π
i
(
S
n
)
⊗
Q
{\displaystyle \pi _{i}(S^{n})\otimes \mathbb {Q} }
for an odd-dimensional sphere (
n
{\displaystyle n}
odd) are zero unless
i
{\displaystyle i}
is equal to 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
{\displaystyle 2n-1}
.
Definition [ edit ]
Let
φ
:
S
2
n
−
1
→
S
n
{\displaystyle \varphi \colon S^{2n-1}\to S^{n}}
be a continuous map (assume
n
>
1
{\displaystyle n>1}
). Then we can form the cell complex
C
φ
=
S
n
∪
φ
D
2
n
,
{\displaystyle C_{\varphi }=S^{n}\cup _{\varphi }D^{2n},}
where
D
2
n
{\displaystyle D^{2n}}
is a
2
n
{\displaystyle 2n}
-dimensional disc attached to
S
n
{\displaystyle S^{n}}
via
φ
{\displaystyle \varphi }
.
The cellular chain groups
C
c
e
l
l
∗
(
C
φ
)
{\displaystyle C_{\mathrm {cell} }^{*}(C_{\varphi })}
are just freely generated on the
i
{\displaystyle i}
-cells in degree
i
{\displaystyle i}
, so they are
Z
{\displaystyle \mathbb {Z} }
in degree 0,
n
{\displaystyle n}
and
2
n
{\displaystyle 2n}
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
{\displaystyle n>1}
), the cohomology is
H
c
e
l
l
i
(
C
φ
)
=
{
Z
i
=
0
,
n
,
2
n
,
0
otherwise
.
{\displaystyle H_{\mathrm {cell} }^{i}(C_{\varphi })={\begin{cases}\mathbb {Z} &i=0,n,2n,\\0&{\text{otherwise}}.\end{cases}}}
Denote the generators of the cohomology groups by
H
n
(
C
φ
)
=
⟨
α
⟩
{\displaystyle H^{n}(C_{\varphi })=\langle \alpha \rangle }
and
H
2
n
(
C
φ
)
=
⟨
β
⟩
.
{\displaystyle H^{2n}(C_{\varphi })=\langle \beta \rangle .}
For dimensional reasons, all cup-products between those classes must be trivial apart from
α
⌣
α
{\displaystyle \alpha \smile \alpha }
. Thus, as a ring , the cohomology is
H
∗
(
C
φ
)
=
Z
[
α
,
β
]
/
⟨
β
⌣
β
=
α
⌣
β
=
0
,
α
⌣
α
=
h
(
φ
)
β
⟩
.
{\displaystyle H^{*}(C_{\varphi })=\mathbb {Z} [\alpha ,\beta ]/\langle \beta \smile \beta =\alpha \smile \beta =0,\alpha \smile \alpha =h(\varphi )\beta \rangle .}
The integer
h
(
φ
)
{\displaystyle h(\varphi )}
is the Hopf invariant of the map
φ
{\displaystyle \varphi }
.
Properties [ edit ]
Theorem : The map
h
:
π
2
n
−
1
(
S
n
)
→
Z
{\displaystyle h\colon \pi _{2n-1}(S^{n})\to \mathbb {Z} }
is a homomorphism.
If
n
{\displaystyle n}
is odd,
h
{\displaystyle h}
is trivial (since
π
2
n
−
1
(
S
n
)
{\displaystyle \pi _{2n-1}(S^{n})}
is torsion).
If
n
{\displaystyle n}
is even, the image of
h
{\displaystyle h}
contains
2
Z
{\displaystyle 2\mathbb {Z} }
. Moreover, the image of the Whitehead product of identity maps equals 2, i. e.
h
(
[
i
n
,
i
n
]
)
=
2
{\displaystyle h([i_{n},i_{n}])=2}
, where
i
n
:
S
n
→
S
n
{\displaystyle i_{n}\colon S^{n}\to S^{n}}
is the identity map and
[
⋅
,
⋅
]
{\displaystyle [\,\cdot \,,\,\cdot \,]}
is the Whitehead product .
The Hopf invariant is
1
{\displaystyle 1}
for the Hopf maps , where
n
=
1
,
2
,
4
,
8
{\displaystyle n=1,2,4,8}
, corresponding to the real division algebras
A
=
R
,
C
,
H
,
O
{\displaystyle \mathbb {A} =\mathbb {R} ,\mathbb {C} ,\mathbb {H} ,\mathbb {O} }
, respectively, and to the fibration
S
(
A
2
)
→
P
A
1
{\displaystyle S(\mathbb {A} ^{2})\to \mathbb {PA} ^{1}}
sending a direction on the sphere to the subspace it spans. It is a theorem, proved first by Frank Adams , and subsequently by Adams and Michael Atiyah with methods of topological K-theory , that these are the only maps with Hopf invariant 1.
Whitehead integral formula [ edit ]
J. H. C. Whitehead has proposed the following integral formula for the Hopf invariant.[2] [3] : prop. 17.22
Given a map
φ
:
S
2
n
−
1
→
S
n
{\displaystyle \varphi \colon S^{2n-1}\to S^{n}}
, one considers a volume form
ω
n
{\displaystyle \omega _{n}}
on
S
n
{\displaystyle S^{n}}
such that
∫
S
n
ω
n
=
1
{\displaystyle \int _{S^{n}}\omega _{n}=1}
.
Since
d
ω
n
=
0
{\displaystyle d\omega _{n}=0}
, the pullback
φ
∗
ω
n
{\displaystyle \varphi ^{*}\omega _{n}}
is a closed differential form :
d
(
φ
∗
ω
n
)
=
φ
∗
(
d
ω
n
)
=
φ
∗
0
=
0
{\displaystyle d(\varphi ^{*}\omega _{n})=\varphi ^{*}(d\omega _{n})=\varphi ^{*}0=0}
.
By Poincaré's lemma it is an exact differential form : there exists an
(
n
−
1
)
{\displaystyle (n-1)}
-form
η
{\displaystyle \eta }
on
S
2
n
−
1
{\displaystyle S^{2n-1}}
such that
d
η
=
φ
∗
ω
n
{\displaystyle d\eta =\varphi ^{*}\omega _{n}}
. The Hopf invariant is then given by
∫
S
2
n
−
1
η
∧
d
η
.
{\displaystyle \int _{S^{2n-1}}\eta \wedge d\eta .}
Generalisations for stable maps [ edit ]
A very general notion of the Hopf invariant can be defined, but it requires a certain amount of homotopy theoretic groundwork:
Let
V
{\displaystyle V}
denote a vector space and
V
∞
{\displaystyle V^{\infty }}
its one-point compactification , i.e.
V
≅
R
k
{\displaystyle V\cong \mathbb {R} ^{k}}
and
V
∞
≅
S
k
{\displaystyle V^{\infty }\cong S^{k}}
for some
k
{\displaystyle k}
.
If
(
X
,
x
0
)
{\displaystyle (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
∞
{\displaystyle V^{\infty }}
, then we can form the wedge products
V
∞
∧
X
.
{\displaystyle V^{\infty }\wedge X.}
Now let
F
:
V
∞
∧
X
→
V
∞
∧
Y
{\displaystyle F\colon V^{\infty }\wedge X\to V^{\infty }\wedge Y}
be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of
F
{\displaystyle F}
is
h
(
F
)
∈
{
X
,
Y
∧
Y
}
Z
2
,
{\displaystyle h(F )\in \{X,Y\wedge Y\}_{\mathbb {Z} _{2}},}
an element of the stable
Z
2
{\displaystyle \mathbb {Z} _{2}}
-equivariant homotopy group of maps from
X
{\displaystyle X}
to
Y
∧
Y
{\displaystyle Y\wedge Y}
. Here "stable" means "stable under suspension", i.e. the direct limit over
V
{\displaystyle V}
(or
k
{\displaystyle k}
, if you will) of the ordinary, equivariant homotopy groups; and the
Z
2
{\displaystyle \mathbb {Z} _{2}}
-action is the trivial action on
X
{\displaystyle X}
and the flipping of the two factors on
Y
∧
Y
{\displaystyle Y\wedge Y}
. If we let
Δ
X
:
X
→
X
∧
X
{\displaystyle \Delta _{X}\colon X\to X\wedge X}
denote the canonical diagonal map and
I
{\displaystyle I}
the identity, then the Hopf invariant is defined by the following:
h
(
F
)
:=
(
F
∧
F
)
(
I
∧
Δ
X
)
−
(
I
∧
Δ
Y
)
(
I
∧
F
)
.
{\displaystyle h(F ):=(F\wedge F)(I\wedge \Delta _{X})-(I\wedge \Delta _{Y})(I\wedge F).}
This map is initially a map from
V
∞
∧
V
∞
∧
X
{\displaystyle V^{\infty }\wedge V^{\infty }\wedge X}
to
V
∞
∧
V
∞
∧
Y
∧
Y
,
{\displaystyle V^{\infty }\wedge V^{\infty }\wedge Y\wedge Y,}
but under the direct limit it becomes the advertised element of the stable homotopy
Z
2
{\displaystyle \mathbb {Z} _{2}}
-equivariant group of maps.
There exists also an unstable version of the Hopf invariant
h
V
(
F
)
{\displaystyle h_{V}(F )}
, for which one must keep track of the vector space
V
{\displaystyle V}
.
References [ edit ]
^ Bott, Raoul; Tu, Loring W (1982). Differential forms in algebraic topology . New York. ISBN 9780387906133 . {{cite book }}
: CS1 maint: location missing publisher (link )
Adams, J. Frank (1960), "On the non-existence of elements of Hopf invariant one", Annals of Mathematics , 72 (1 ): 20–104, CiteSeerX 10.1.1.299.4490 , doi :10.2307/1970147 , JSTOR 1970147 , MR 0141119
Adams, J. Frank ; Atiyah, Michael F. (1966), "K-Theory and the Hopf Invariant", Quarterly Journal of Mathematics , 17 (1 ): 31–38, doi :10.1093/qmath/17.1.31 , MR 0198460
Crabb, Michael; Ranicki, Andrew (2006). "The geometric Hopf invariant" (PDF) .
Hopf, Heinz (1931), "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche", Mathematische Annalen , 104 : 637–665, doi :10.1007/BF01457962 , ISSN 0025-5831
Shokurov, A.V. (2001) [1994], "Hopf invariant" , Encyclopedia of Mathematics , EMS Press
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Hopf_invariant&oldid=1222384397 "
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