3.1 Projective Lines

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
. We
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

where has

These nontrivial projections all have rank 1, so they are the points of our would-be projective space. Our would-be projective space has just one line, corresponding to the projection 1, and all the points lie on this line. It is easy to check that the axioms for a projective space hold. Since this projective space is 1-dimensional, we have succeeded in creating the

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

We call an equivalence class for this relation a

Be careful: when is the octonions, the line through the
origin containing is not always equal to

This is only true when is associative, or when or is . Luckily, we have when and when . Thus in either case we get a concrete description of the line through the origin containing : when it equals

and when it equals

In particular, the line through the origin containing is always a real vector space isomorphic 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

is given by stereographic projection:

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

misses only the south pole. Composing the first map with the inverse of the second, we get the map , which goes by the name of 'conformal inversion'. The southern hemisphere of the Riemann sphere consists of all points with , while the northern hemisphere consists of all with . Unit complex numbers give points on the equator.

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 :

We can think of it as the one-point compactification of . The 'southern hemisphere', 'northern hemisphere', and 'equator' of have descriptions exactly like those given above for the complex case. Also, as in the complex case, the maps and are angle-preserving with respect to the usual Euclidean metric on and the round metric on the sphere.

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

What are these maps? We can describe them all simultaneously. Suppose is a normed division algebra of dimension . In the southern hemisphere of , we can identify any fiber of with by mapping the point in the line to the element . This trivializes the canonical line bundle over the southern hemisphere. Similarly, we can trivialize this bundle over the northern hemisphere by mapping the point in the line to the element . If has norm one, is a point on the equator, so we get two trivializations of the fiber over this point. These are related as follows: if then . The map is thus right multiplication by . In short,

is just the map sending any norm-one element to the operation of right multiplication by .

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:

- generates .
- generates .
- generates .
- generates .

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**:

The projection is called the

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,
and let
be any smooth map. If is
the normalized volume form on , then is a closed
-form on . Since the th cohomology of
vanishes,
for some -form .
We define the **Hopf invariant** of to be the number

This is easily seen to be invariant under smooth homotopies of the map .

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