A one-dimensional projective space is called a projective line. Projective lines are not very interesting from the viewpoint of axiomatic projective geometry, since they have only one line on which all the points lie. Nonetheless, they can be geometrically and topologically interesting. This is especially true of the octonionic projective line. As we shall see, this space has a deep connection to Bott periodicity, and also to the Lorentzian geometry of 10-dimensional spacetime.
Suppose is a normed division algebra. We have already defined
when is associative, but this definition does not work well
for the octonions: it is wiser to take a detour through Jordan
algebras. Let be the space of hermitian
matrices with entries in . It is easy to check that this becomes a
Jordan algebra with the product
can try to build a projective space from this Jordan algebra using the
construction in the previous section. To see if this
succeeds, we need to ponder the projections in . A little
calculation shows that besides the trivial projections 0 and 1, they
are all of the form
Given any nonzero element
, we can normalize it and then
use the above formula to get a point in , which we call
. This allows us to describe in terms
of lines through the origin, as follows. Define an equivalence relation
on nonzero elements of by
Be careful: when is the octonions, the line through the
origin containing is not always equal to
We can make into a manifold as follows. By the above observations, we can cover it with two coordinate charts: one containing all points of the form , the other containing all points of the form . It is easy to check that iff , so the transition function from the first chart to the second is the map . Since this transition function and its inverse are smooth on the intersection of the two charts, becomes a smooth manifold.
When pondering the geometry of projective lines it is handy to
visualize the complex case, since is just the familiar
'Riemann sphere'. In this case, the map
where we choose the sphere to have diameter 1. This map from to
is one-to-one and almost onto, missing only the point at
infinity, or 'north pole'. Similarly, the map
All these ideas painlessly generalize to for any normed division
algebra . First of all, as a smooth manifold is just a
sphere with dimension equal to that of :
One of the nice things about is that it comes equipped with a vector bundle whose fiber over the point is the line through the origin corresponding to this point. This bundle is called the canonical line bundle, . Of course, when we are working with a particular division algebra, 'line' means a copy of this division algebra, so if we think of them as real vector bundles, and have dimensions 1,2,4, and 8, respectively.
These bundles play an important role in topology, so it is good to
understand them in a number of ways. In general, any -dimensional
real vector bundle over can be formed by taking trivial bundles
over the northern and southern hemispheres and gluing them together
along the equator via a map
. We must
therefore be able to build the canonical line bundles
and using maps
The importance of the map becomes clearest if we form the inductive limit of the groups using the obvious inclusions , obtaining a topological group called . Since is included in , we can think of as a map from to . Its homotopy class has the following marvelous property, mentioned in the Introduction:
Another nice perspective on the canonical line bundles comes from
looking at their unit sphere bundles. Any fiber of is naturally
an inner product space, since it is a line through the origin in .
If we take the unit sphere in each fiber, we get a bundle of
-spheres over called the Hopf bundle:
We can understand the Hopf maps better by thinking about inverse images of points. The inverse image of any point is a -sphere in , and the inverse image of any pair of distinct points is a pair of linked spheres of this sort. When we get linked circles in , which form the famous Hopf link:
When , we get a pair of linked 7-spheres in .
To quantify this notion of linking, we can use the 'Hopf invariant'.
Suppose for a moment that is any natural number greater than one,
be any smooth map. If is
the normalized volume form on , then is a closed
-form on . Since the th cohomology of
for some -form .
We define the Hopf invariant of to be the number
To see how the Hopf invariant is related to linking, we can compute it using homology rather than cohomology. If we take any two regular values of , say and , the inverse images of these points are compact oriented -dimensional submanifolds of . We can always find an oriented -dimensional submanifold that has boundary equal to and that intersects transversely. The dimensions of and add up to , so their intersection number is well-defined. By the duality between homology and cohomology, this number equals the Hopf invariant . This shows that the Hopf invariant is an integer. Moreover, it shows that when the Hopf invariant is nonzero, the inverse images of and are linked.
Using either of these approaches we can compute the Hopf invariant of , and . They all turn out to have Hopf invariant 1. This implies, for example, that the inverse images of distinct points under are nontrivially linked 7-spheres in . It also implies that , and give nontrivial elements of for , and . In fact, these elements generate the torsion-free part of .
A deep study of the Hopf invariant is one way to prove that any division algebra must have dimension 1, 2, 4 or 8. One can show that if there exists an -dimensional division algebra, then must be parallelizable: it must admit pointwise linearly independent smooth vector fields. One can also show that for , is parallelizable iff there exists a map with . The hard part is the final step: showing that a map from to can have Hopf invariant 1 only if , or . Proving this requires algebraic topology that goes far beyond the scope of this paper [2,9,52]. There is just one thing we wish to note about this proof: it involves Bott periodicity, which we describe in the next section. As we shall see, Bott periodicity has a natural explanation in terms of the canonical line bundle over . There is thus a sense in which the existence of the octonions is 'responsible' for the nonexistence of division algebras in dimensions other than 1, 2, 4, and 8!
© 2001 John Baez