One way of obtaining K-theory classes is to take a -term complex of vector bundles
we get get a K-theory class
. In fact all K-theory classes occur like this if we allow
to be infinite dimensional. This motivates the study of Fredholm operators.
Fix to be a separable Hilbert space over . [A vector space with an inner product that induces a complete metric on V].
Let be a bounded linear map (the image of the unit ball is bounded).
- We say that is Fredholm if both and are finite dimensional.
- The index of a Fredolm operator is defined .
- Fredholm operators .
Consider the Hilbert space . The maps
In fact is a left inverse of . The operator is the finite rank operator:
Constructing Index bundles[edit | edit source]
The theory of Fredholm operators is closely related to K-theory: Suppose that we had a smoothly varying family of Fredholm operators over a compact, connected space . Then such a family gives a K-theory class in , and all K-theory classes arise in this way.
Let be a family of Fredholm operators . Then we may associate a canonically defined K-theory class .
The map is continuous so is constant. In the simplified situation where and hence are also constant we can give an explicit description of the index bundles.
inherit the structure of vector bundles from the family. Define
Now for the general situation consider the map (assuming for simplicity that ). Fredholm operators are invertible up to compact operators, so for each . By compactness of there is a global choice of that works for all . Now the family has Cokernel of constant dimension so apply the above construction. [HARD BIT: prove this is independent of ].
Show that all K(X) classes arise this way by considering the sequence
is the trivial vector bundle of
. The second map is
Fredholm operators and winding numbers.
Let be the space of square integrable (complex) functions on . All such functions have a Fourier series
Define the Hardy space
to be the subspace of functions with
Let be a continuous function. Define an operator . Note that for . Define to be the projection of to by killing the negative Fourier coefficients.
is Fredholm . The nice fact about these Fredholm operators is that the index is the winding number of around .