Let be a vector bundle over , and let be a continuous map. There’s an induced bundle over , given by (what’s the projection map?). Suppose is a vector bundle; the following result states that this is classified by a map from to some space.
Lemma: There’s a universal bundle, denoted , over a space , which induces , such that any rank vector bundle over a paracompact space is induced by a map , and two such bundles are isomorphic if they’re induced by homotopic maps.
Proof. Let denote the space of -dimensional complex linear subspaces of . The open subsets of are sets of subspaces which intersect an open subset of . We can define a vector bundle over via . If is induced by the inclusion , then . Thus we can consider the union of under the inclusion maps to define the classifying space of the unitary group . QED.
Let’s consider the inclusions , and denote by its direct limit. The classifying space of is denoted . This is also the direct limit of the inclusions . It’s easy to see that a general vector bundle over corresponds via the above important result to a map , and this vector bundle is of rank if is the minimal integer such that the map factors as .
Definition: Let be a fixed base space that is compact and Hausdorff. Denote by the Grothendieck group of of isomorphism classes of finite-dimensional complex vector bundles over with addition given by and multiplication given by . Denote by the kernel of the map , induced by the inclusion . This is called reduced K-theory. There is an analogous definition of using real vector bundles.
Example: If , then and thus .
Before going to the next example, we will prove the following lemma from linear algebra.
Lemma: deformation retracts onto .
Proof. Let , and consider the QR decomposition where is unitary and is a upper triangular matrix. Define as:
What follows is a more nontrivial computation, namely that of . I learnt this from Professor Miller’s lecture notes (http://www.math.harvard.edu/~ecp/latex/misc/haynes-notes/haynes-notes.pdf). We can write as two disks glued at the equator (which is ). The disks are contractible and hence any vector bundle is trivial, i.e., is of the form . Any vector bundle on can be trivialized on the hemispheres, and thus are completely determined by choosing a linear isomorphism in a continuous manner. In other words, . The map associated to a complex -dimensional vector bundle is called the clutching function. This statement is true even if we place with for some (and replace with ).
By the lemma, deformation retracts onto . Now, acts on , and the stabilizer of this action is . The map given by for some fixed basepoint is a fibration with fibers . Consider the long exact sequence in homotopy groups .
This shows that for . Now, , so . An element therefore corresponds to the clutching function for a complex line bundle . In particular, the universal line bundle over corresponds to the clutching function . The inclusion shows that the element corresponds to the clutching function of the bundle . Thus . We will now prove an intermediary result.
Lemma: Let denote the universal line bundle over . Then .
Proof. By our above analysis, it suffices to show that the respective clutching functions and are homotopic. We know that is path-connected. There is therefore a path, say , from to the matrix that flips the factors in . The composition is a homotopy from to where for . (When , this is , and when , this is .) QED.
Finally, this means that:
Result: , and .
Another example: recall the statement . Suppose . Then , but is connected, so .
Recall the following result from https://erdosninth.wordpress.com/2016/04/17/a-long-exact-sequence-in-topological-k-theory/, which hints to us that K-theory might be some kind of cohomology theory.
Theorem: Let be a closed subset of a compact Hausdorff space . There’s a long exact sequence: .
Another important result:
Theorem (Bott periodicity): There’s an isomorphism .
Proof. Omitted. Maybe I should include this? It’s similar to the computation of .
Definition: Define for , and for some such that .
The form a generalized cohomology theory, called (reduced) complex K-theory. Unreduced complex K-theory can also be defined. In fact, where is as defined above. Brown representability tells us that complex K-theory is represented by a spectrum . We have already shown that . It’s easy to determine .
Complex K-theory is a multiplicative cohomology theory. This comes from the following H-space structure on . Define maps that is characterized up to homotopy as where the are the universal -bundles over .
For example, since , it follows that:
Recall that is generated by (where we just write for ). This is isomorphic to , and the generator of is denoted and called the Bott element. It follows that as a graded ring, .
Also, recall that one can define via real vector bundles. One can run through the same constructions as above, and find that:
- There are spaces constructed as the direct limit of the (defined as the space of -dimensional linear subspaces of ) such that .
- The real K-theory of the -sphere is .
- There is an isomorphism . Define for , and for some such that .
- Let . Then is a generalized cohomology theory, represented by a spectrum with . Note that Bott periodicity for says that .
It turns out that is much more complicated. There is an isomorphism of graded rings where , , and .
One might wonder about the relationship between and — after all, both are “K-theories” defined using vector bundles! Complexification of vector bundles does indeed give a multiplicative homomorphism , and realification gives a homomorphism of -modules . On homotopy groups (cf. Rognes’ notes on topological K-theory), complexification gives a map sending , , and . Realification gives a map sending .
Next time we’ll finish the proof of the Hopf invariant one problem. The way we’ll do this is by redefining the Hopf invariant using K-theory, and proving that our new definition is equivalent to the definition we already have. Then we’ll have to do a little more analysis. This part of the series might be a bit delayed since final exams are coming up in weeks.