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
and proved that
is essential, i.e., not
homotopic to the constant map, by using the fact that the linking number of the circles
is equal to 1, for any
.
is the infinite
cyclic group generated by
. In 1951,
Jean-Pierre Serre proved that the
rational homotopy groups
[1]
for an odd-dimensional sphere (
odd) are zero unless
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
.
Definition
Let
be a
continuous map (assume
). Then we can form the
cell complexC\varphi=Sn\cup\varphiD2n,
where
is a
-dimensional disc attached to
via
.The cellular chain groups
are just freely generated on the
-cells in degree
, so they are
in degree 0,
and
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
), the cohomology is
\varphi)=\begin{cases}Z&i=0,n,2n,\ 0&otherwise.\end{cases}
Denote the generators of the cohomology groups by
and
H2n(C\varphi)=\langle\beta\rangle.
For dimensional reasons, all cup-products between those classes must be trivial apart from
. Thus, as a
ring, the cohomology is
=Z[\alpha,\beta]/\langle\beta\smile\beta=\alpha\smile\beta=0,\alpha\smile\alpha=h(\varphi)\beta\rangle.
The integer
is the
Hopf invariant of the map
.
Properties
Theorem: The map
is a homomorphism. If
is odd,
is trivial (since
is torsion).If
is even, the image of
contains
. Moreover, the image of the
Whitehead product of identity maps equals 2, i. e.
, where
is the identity map and
is the
Whitehead product.
The Hopf invariant is
for the
Hopf maps, where
, corresponding to the real division algebras
, respectively, and to the fibration
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
, one considers a
volume form
on
such that
.Since
, the
pullback
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
-form
on
such that
. The Hopf invariant is then given by
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
denote a vector space and
its
one-point compactification, i.e.
and
for some
.
If
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
, then we can form the wedge products
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
is
an element of the stable
-equivariant homotopy group of maps from
to
. Here "stable" means "stable under suspension", i.e. the direct limit over
(or
, if you will) of the ordinary, equivariant homotopy groups; and the
-action is the trivial action on
and the flipping of the two factors on
. If we let
\DeltaX\colonX\toX\wedgeX
denote the canonical diagonal map and
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
-equivariant group of maps.There exists also an unstable version of the Hopf invariant
, for which one must keep track of the vector space
.
References
Notes and References
- 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 .
- 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 .
- Book: Bott . Raoul . Tu . Loring W . Differential forms in algebraic topology . 1982 . New York . 9780387906133.