• Source: Hopf invariant

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.




Motivation


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





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


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


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


J. H. C. Whitehead has proposed the following integral formula for the Hopf invariant.: 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


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



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

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