2 The spaces and
A better way to think about Bott periodicity is to not look at , but . To describe , we again start with the “unstable” versions .
is defined to be a space such that for any CW complex , there is a canonical bijection
An explicit model of can be described as the Grassmannian of -planes in , the countable dimension complex vector space.
This universal property of is very useful because it gives us a very geometric handle on the spaces . For example, the direct sum and tensor product of vector bundles are classified by maps
The first question to ask is — how does relate to ? Fix any base point of , and consider the space of based loops in , written .
The core content of the statement is that clutching functions work. Indeed, suppose
is a connected based space. Then we have
where the last equality comes from (e.g. by inspecting the construction of as a Grassmannian to see it only has cells of dimension ). So the proposition is equivalent to saying that vector bundles on are the same as (based) maps , which is exactly the clutching construction.
The importance of this proposition is that it allows us to read off the homotopy groups of from those of . Of course, this is not too useful until we pass on to the limit . There is a map given by adding a trivial line bundle. Under the clutching construction, this corresponds to the map we had previously. We then let
In particular, there is a map which we will choose to be our canonical basepoint of .
The direct sum and of vector bundles is compatible with the inclusion , and so gives rise to a map
We would like a map that comes from tensor products as well, but that is not compatible with the inclusion, since
To fix this, we need to think about what represents.
A virtual vector bundle is a formal difference of two vector bundles.
More precisely, if is a finite CW complex, write for the monoid of vector bundles over (up to isomorphism) under direct sum. Write to be the group completion of . A virtual vector bundle is then an element of .
is a vector bundle, we write
for its image in
. Then every element in
is of the form
, and its rank
. We also write
-dimensional trivial vector bundle.
For any vector bundle over , there is some other vector bundle such that is trivial.
Hence, any virtual vector bundle can be written as for some vector bundle .
If is a finite CW complex, then is the group of rank virtual vector bundles, where the group structure comes from the direct sum map .
is finite, we have
If a map classifies a vector bundle , then the correspondence sends this to .
If is a finite CW complex, then
factor to keep track of the rank, since every virtual vector bundle is the sum of a rank zero virtual vector bundle plus a trivial bundle.
Now since is linear, it induces a map , classified by a map
In fact, we get something even better, since the basepoint of , corresponding to the trivial rank 0 vector bundle, kills everything under , so this factors to give a map
where as always, .
This is important, since it induces a ring structure on — if , then smashing them together gives
As a group, the ring is in every even degree, and is zero otherwise. The ring structure is the best you can hope for.
(Complex Bott periodicity)
This has some nice geometric consequences. Observe that abstractly as groups, and we know this without using the ring structure. This does not automatically imply , since we need a map that realizes this isomorphism of groups in order to apply Whitehead's theorem. The ring structure provides exactly this.
Indeed, let be a generator of . Then we get a map
The adjoint map is then multiplication by , which is an isomorphism. So
The map above gives a homotopy equivalence
This is a geometric
incarnation of the Bott periodicity theorem, which says two spaces
are homotopy equivalent.
Given the importance of the map , it is reassuring to know there is a very concrete description of it:
The class can be chosen to be represented by the map
Equivalently, it is , where is the tautological bundle over .
The real version of these results is slightly less pretty.
(Real Bott periodicity)
where . Therefore,