Hopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.
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 linking number (=1) of the circles
for any
.
It was later shown that the homotopy group is the infinite cyclic group generated by
. In 1951, Jean-Pierre Serre proved that the rational homotopy groups
for an odd-dimensional sphere ( 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
.
Definition
Let be a continuous map (assume
). Then we can form the cell complex
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
Denote the generators of the cohomology groups by
-
and
For dimensional reasons, all cup-products between those classes must be trivial apart from . Thus, as a ring, the cohomology is
The integer is the Hopf invariant of the map
.
Properties
Theorem: is a homomorphism. Moreover, if
is even,
maps onto
.
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 Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1.
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
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
denote the canonical diagonal map and the identity, then the Hopf invariant is defined by the following:
This map is initially a map from
to
,
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
- Adams, J.F. (1960), "On the non-existence of elements of Hopf invariant one", Ann. Math. (The Annals of Mathematics, Vol. 72, No. 1) 72 (1): 20–104, doi:10.2307/1970147, JSTOR 1970147
- Adams, J.F.; Atiyah, M.F. (1966), "K-Theory and the Hopf Invariant", The Quarterly Journal of Mathematics 17 (1): 31–38, doi:10.1093/qmath/17.1.31
- Crabb, M.; Ranicki, A. (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), "Hopf invariant", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4