Talbot Workshop 2010 Talk 2: K-theory and Index Theory
From the point of view of an analyst, one of the most delightful things about complex K-theory is that it has a nice realization by analytical objects, namely (pseudo)differential operators and their Fredholm indices. This connection allows quite a bit of interesting information to flow both ways: from analysis to topology and vice versa.
This talk will try and give a sketch of this picture, and consists of three parts or themes. The first is “the Gysin map as the index,” describing the families index theorem of Atiyah and Singer, and how the pushforward along a fibration in K-theory can be realized as the index of a family of operators. The second is “spin as an orientation,” in which I discuss Clifford algebras, spin and spin structures, Dirac operators and the analytic realization of the Thom isomorphism for complex K-theory. Finally (I did not have time to get to this part in the actual Talbot talk) I will discuss the constructions leading to higher index maps (i.e. instead of ; of course this is more interesting for real K-theory than for complex K-theory), namely Clifford-linear differential and Fredholm operators. The best reference for almost everything in this talk is the wonderful book [lawson1989spin] by Lawson and Michelson. I cannot recommend this book highly enough. I will also try and give references to original sources (essentially all of which involve Atiyah as an author).
1. Some notation and facts
First let us get down some notation and facts about K-theory that will be of use in the following. Let be a (not necessarily complex) vector bundle. There are many ways of constructing the Thom space of , which will be denoted :
The first space denotes the unit disk bundle of (with respect to some choice of metric) quotiented out by the unit sphere bundle; the second denotes the radial compactification of quotiented out by its boundary; the third denotes the projective bundle111Note that this constructs the fiberwise one point compactification of . associated to (where is the trivial complex line bundle), modulo the section at infinity. Indeed we could take any compactification of modulo the points added at infinity.
In any cohomology theory, the reduced cohomology of is of interest. In index theory, however, we prefer to think of the K-theory of as compactly supported K-theory222Though we shall only need it for vector bundles, by compactly supported cohomology of any space can be thought of as the relative cohomology of where denotes the (one-point or otherwise) compactification of . This is consistent if we agree to take for compact . of the space :
There is a convenient representation of relative even K-theory of a pair , where is a nice enough subset, known as the difference bundle construction:
where are vector bundles and is a bundle map covering the identity on , which restricts to an isomorphism over . The equivalence relations amount to stabilization and homotopy. Intuitively this should be clear; if and are isomorphic over , then, should be trivial in K-theory when restricted to .
Unpacking this in the case of compactly supported K-theory for , we conclude that we can represent by
where is the projection and denotes the zero section. Indeed, the pair is homotopy equivalent to , and by contractibility of the fibers of , any vector bundles over are homotopic to ones pulled up from the base, i.e. of the form .
Let now be a complex vector bundle. The Thom isomorphism in K-theory states that has a K-theory orientation, so that . Specifically, is a freely generated, rank one module over , and in the representation of compactly supported K-theory discussed above, the generator, or Thom class333We’ll discuss orientation classes more generally in section 3, and we’ll interpret the Thom class in terms of spin structures in section LABEL:S:structures. has the following nice description.
The Thom class for a complex vector bundle can be represented as the element
where the isomorphism off is given by , the first term denoting exterior product with and the second denoting the contraction with (equivalently, the inner product with ), with respect to any choice of metric.
The isomorphism is an example of Clifford multiplication, about which we will have much more to say in section LABEL:S:clifford.
Finally, a bit about Fredholm operators. Let be a separable, infinite dimensional Hilbert space, and recall that a bounded linear operator is Fredholm if it is invertible modulo compact operators, which in turn are those operators in the norm closure of the finite rank operators. Thus is Fredholm iff there exists an operator such that and are compact. One upshot of this is that
The relationship between Fredholm operators and K-theory starts with the observation of Atiyah [atiyah1967ktheory] that the space of Fredholm operators on classifies :
where the left hand side denotes homotopy classes of maps , the latter given the operator topology.444The precise topology one should take on becomes a little difficult in twisted K-theory, but (I guess!) not here.
Morally, the idea is to take a map , and examine the vector bundles and , whose fibers at a point are the finite dimensional vector spaces and , respectively. Of course this is a bit of a lie, since the ranks of these bundles will generally jump around as varies; nevertheless, it is possible to stabilize the situation and see that the class
2. Differential operators and families
One of the most important sources of such maps are families of differential operators on . Let’s start with differential operators themselves. A working definition of the differential operators of order , , where and are vector bundles over is the following local definition.
where we’re employing multi-index notation: , , , where are local coordinates on .
This local expression for does not transform well under changes of coordinates; however, the highest order terms (those with ) do behave well. If we consider , we can view it as a monomial map of order . If are coordinates on inducing coordinates on , the monomial obtained is just
Summing up all the terms of order gives us a homogeneous polynomial of order , which because of the term is a homogeneous polynomial on valued in . The claim is that this principal symbol
An operator is elliptic if its principal symbol is invertible away from the zero section . The canonical example of an elliptic operator is , the Laplacian (on functions, say), a second order operator whose principal symbol is , where and the norm comes from a Riemannian metric. The canonical non-example on is , the D’Alembertian or wave-operator, whose principal symbol is , where , which vanishes on the (light) cones .
The reader whose was paying particularly close attention earlier will note that the symbol of an elliptic differential operator is just the right kind of object to represent an element in the compactly supported K-theory555In fact, any element of can be represented as the symbol of an elliptic pseudodifferential operator, though we shall not discuss these here. of since it is invertible away from :
For our purposes, the other important feature of an elliptic operator on a compact manifold is that it extends to a Fredholm operator . Actually, this is a bit of a lie, since if , is unbounded on , and we should really consider it acting on its maximal domain in , which is the Sobolev space which itself has a natural Hilbert space structure. However, since the order of operators is immaterial as far as index theory is concerned, we will completely ignore this issue for the rest of this note, pretending all operators in sight are of order zero666In fact it is always possible to compose with an invertible pseudodifferential operator (of order ) so that the composite has order zero, without altering the index of . One such choice is , another is ., which act boundedly on .
Now let be a fibration of compact manifolds with fibers . A family of differential operators with respect to , is just a set of differential operators on (vector bundles over) the fibers , parametrized smoothly by the base . For a formal definition, take the principal bundle such that ; then the differential operator families of order are obtained as777I will be sloppy about distinguishing between families vector bundles on the fibers and vector bundles on the total space . In fact they are the same.
As before, there is a principal symbol map
where denotes the vertical (a.k.a. fiber) cotangent bundle. Once again is elliptic if is invertible away from the zero section; if this is the case, extends to a family of Fredholm operators on the Hilbert space bundles
By Kuiper’s theorem that the unitary group of an infinite dimensional Hilbert space is contractible, the bundles are trivializable, so trivializing and identifying and (all separable, infinite dimensional Hilbert spaces are isomorphic), we obtain a map
which must therefore have an index in the even K-theory of :
Just as in the case of the single operator, the principal symbol of the family represents a class in compactly supported K-theory of :
We will come back to the relationship between these two objects in a moment; for now you should think of the index as an assignment which maps to . This is well-defined since any two elliptic operators with the same principal symbol are homotopic through elliptic (hence Fredholm) operators, and since the index is homotopy invariant, any two choices of operators with the same symbol will have the same index in .
3. Gysin maps
In the first Talbot talk, Jesse Wolfson discussed the Gysin map in K-theory associated to an embedding. We will need a similar kind of Gysin map associated to fibrations. Let be a smooth fibration with compact but not necessarily having compact fibers. By the theorem of Whitney, we can embed any manifold into for sufficiently large , and since is compact, this can be done fiberwise to obtain an embedding of fibrations from into a trivial fibration: