Hopf invariant explained

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

Motivation

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map

η\colonS3\toS2,

and proved that

η

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)\subsetS3

is equal to 1, for any

xy\inS2

.
2)
\pi
3(S
is the infinite cyclic group generated by

η

. In 1951, Jean-Pierre Serre proved that the rational homotopy groups [1]
n)
\pi
i(S

Q

for an odd-dimensional sphere (

n

odd) are zero unless

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

2n-1

.

Definition

Let

\varphi\colonS2n-1\toSn

be a continuous map (assume

n>1

). Then we can form the cell complex

C\varphi=Sn\cup\varphiD2n,

where

D2n

is a

2n

-dimensional disc attached to

Sn

via

\varphi

.The cellular chain groups
*
C
cell(C

\varphi)

are just freely generated on the

i

-cells in degree

i

, so they are

Z

in degree 0,

n

and

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

), the cohomology is
i
H
cell(C

\varphi)=\begin{cases}Z&i=0,n,2n,\ 0&otherwise.\end{cases}

Denote the generators of the cohomology groups by

n(C
H
\varphi)

=\langle\alpha\rangle

and

H2n(C\varphi)=\langle\beta\rangle.

For dimensional reasons, all cup-products between those classes must be trivial apart from

\alpha\smile\alpha

. Thus, as a ring, the cohomology is
*(C
H
\varphi)

=Z[\alpha,\beta]/\langle\beta\smile\beta=\alpha\smile\beta=0,\alpha\smile\alpha=h(\varphi)\beta\rangle.

The integer

h(\varphi)

is the Hopf invariant of the map

\varphi

.

Properties

Theorem: The map

h\colon\pi2n-1(Sn)\toZ

is a homomorphism. If

n

is odd,

h

is trivial (since

\pi2n-1(Sn)

is torsion).If

n

is even, the image of

h

contains

2Z

. Moreover, the image of the Whitehead product of identity maps equals 2, i. e.

h([in,in])=2

, where

in\colonSn\toSn

is the identity map and

[,]

is the Whitehead product.

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(A2)\toPA1

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.[2] [3] Given a map

\varphi\colonS2n-1\toSn

, one considers a volume form

\omegan

on

Sn

such that
\int
Sn

\omegan=1

.Since

d\omegan=0

, the pullback

\varphi*\omegan

is a closed differential form:

d(\varphi*\omegan)=\varphi*(d\omegan)=\varphi*0=0

.By Poincaré's lemma it is an exact differential form: there exists an

(n-1)

-form

η

on

S2n

such that

dη=\varphi*\omegan

. The Hopf invariant is then given by
\int
S2n

η\wedgedη.

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

denote a vector space and

Vinfty

its one-point compactification, i.e.

V\congRk

and

Vinfty\congSk

for some

k

.

If

(X,x0)

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

Vinfty

, then we can form the wedge products

Vinfty\wedgeX.

Now let

F\colonVinfty\wedgeX\toVinfty\wedgeY

be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of

F

is

h(F)\in\{X,Y\wedge

Y\}
Z2

,

an element of the stable

Z2

-equivariant homotopy group of maps from

X

to

Y\wedgeY

. 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

Z2

-action is the trivial action on

X

and the flipping of the two factors on

Y\wedgeY

. If we let

\DeltaX\colonX\toX\wedgeX

denote the canonical diagonal map and

I

the identity, then the Hopf invariant is defined by the following:

h(F):=(F\wedgeF)(I\wedge\DeltaX)-(I\wedge\DeltaY)(I\wedgeF).

This map is initially a map from

Vinfty\wedgeVinfty\wedgeX

to

Vinfty\wedgeVinfty\wedgeY\wedgeY,

but under the direct limit it becomes the advertised element of the stable homotopy

Z2

-equivariant group of maps.There exists also an unstable version of the Hopf invariant

hV(F)

, for which one must keep track of the vector space

V

.

References

Notes and References

  1. Serre . Jean-Pierre . Groupes D'Homotopie Et Classes De Groupes Abeliens . The Annals of Mathematics . September 1953 . 58 . 2 . 258–294 . 10.2307/1969789. 1969789 .
  2. Whitehead . J. H. C. . An Expression of Hopf's Invariant as an Integral . Proceedings of the National Academy of Sciences . 1 May 1947 . 33 . 5 . 117–123 . 10.1073/pnas.33.5.117. 16578254 . free . 1079004 . 1947PNAS...33..117W .
  3. Book: Bott . Raoul . Tu . Loring W . Differential forms in algebraic topology . 1982 . New York . 9780387906133.