Can one use Atiyah-Singer to prove that the Chern-Weil definition of Chern classes are $\mathbb{Z}$-cohomology classes?

Question

In Chern-Weil theory, we choose an arbitrary connection $\nabla$ on a complex vector bundle $E\rightarrow X$, obtain its curvature $F_\nabla$, and then we get Chern classes of $E$ from the curvature form. A priori it looks like these live in $H^*(X;\mathbb{C})$, but by an argument that I don't really understand they're actually in the image of $H^*(X;\mathbb{Z})$ (which is where they're usually considered to live). Meanwhile, I've heard people say that whenever I see a arbitrary real constants that end up having to be integers I should wonder whether the Atiyah-Singer index theorem is lurking in there somewhere. Is there anything to this wild guess?

Answer 1

It's true that in some sense the Atiyah-Singer Index Theorem has led to some integrality results. The theorem states that for an elliptic operator on a compact manifold two numbers are equal. One of them, the "analytic index", is obviously an integer. The other one, the "topological index", which depends on the symbol of the operator, is obviously a rational number but not so obviously an integer.

On the other hand, you can argue that the integrality phenomena revealed in this way don't really have anything to do with the analysis.

There are two equivalent ways to define the topological index. There is the cohomological formula in terms of characteristic classes that live in rational cohomology. There is also the conceptually more fundamental $K$-theory definition of the topological index, in which you let the symbol of the operator give you an element of the compactly supported $K^0$ of the cotangent manifold and then use Bott periodicity and the fact that $K$-theory is a cohomology theory to get from here to an element of $K^0(point)\cong\mathbb Z$. The fact that the integer thus defined coincides with the rational number obtained from characteristic classes is a calculation involving the relationship between $K$-theory and ordinary cohomology, and does not rely on the fact that there was a differential operator involved.

Answer 2

The integrality results coming out of the index theorem and that for the Chern-Weil classes have a different flavour.

The Chern-Weil classes are integral cohomology classes for any vector bundle on any manifold.

On the other hand, contemplate on, say, the Hirzebruch signature theorem (as a typical example for the integrality results coming from the index theorem). It implies that if $M$ is a closed oriented $4k$-manifold, then a well-known rational polynomial in the Pontrajgin classes of the tangent bundle in degree $4k$ is in integral cohomology. The statement is about very special bundles (tangent bundles) on very special manifolds, and it is only about the component in the top degree and it is easy to see that the statement fails for other vector bundles or away from the top degree.

I do not think you can derive the integrality of Chern classes from the index theorem (or from the less analytic integrality result that Tom Goodwillie describes).

Answer 3

The usual argument (that you had trouble with) is not so complicated: you prove that every vector bundle is the pullback of the ``universal'' vector bundle on a Grassmannian. Then you compute the Chern classes of the universal bundle, directly on the Grassmannian, and by a miracle they turn out to be integral. In fact, you can compute the relevant integrals explicitly by hand, so it turns out quite nicely. So the hard bit is showing that every vector bundle is the pullback of the universal bundle; see Milnor and Stasheff, p. 65. As for Atiyah-Singer, that seems much more complicated and I don't see how to make use of it here.

Answer 4

My first remark about this question is a little bit pithy - the standard cohomological index formula works only for even dimensional manifolds while Chern-Weil theory is of course more general.

For a more substantive answer, let's look at the index formula itself:

$Index(D) = \int_{T^*M} ch(\sigma_D) Todd(TM \otimes \mathbb{C})$

Here $ch(\sigma_D)$ refers to the chern character of the "symbol class" in K-theory of the elliptic operator $D$. My first observation is that this formula places special emphasis on the Todd class of the tangent bundle, and it seems to me that one would have to place similar emphasis on the Todd class in Chern - Weil theory in order to answer your question in the affirmative. I certainly do not claim that this is impossible - the Todd polynomial seems to take a privileged position in many places where it does not seem to belong - but I do not know how to do it.

Additionally, we are forced to ask what characteristic classes can be written in the form $ch(\sigma_D)$. As Tom Goodwillie points out, we will get an integrality result if we plug in any K-theory class $x$ rather than a K-theory class of the form $\sigma_D$ (although it may be that all classes in $K(T^*M)$ are symbol classes), but there are still constraints on which characteristic classes can appear. For example, we will only get classes in cohomology of even degree.

So there are a number of hints that the right hand side of the index formula is too specialized for Chern-Weil theory. But the left hand side also helps answer your question. Here are two general principles in index theory that will help me explain:

1. The index theorem for any operator can be deduced from the index theorem for a special class of operators on a special class of manifolds - the "spin$^c$ Dirac operators". These operators are completely determined by the geometry of the manifolds on which they live.
2. The index depends continuously on the operator in an appropriate sense. In particular, if $D_t$ is a continuous family of operators then $Index(D_t)$ is constant in $t$.

Combining these two facts, we see that the integers which we can obtain from the index theorem depend only on homotopy theoretic information about the manifold and the coefficient bundle (if applicable). This is not true of all integers that can be obtained via Chern-Weil theory, and it demonstrates that index theoretic invariants are much too crude for ordinary cohomology theory.