May 29, 2012
I’d like to take a break from the previous homotopy-theoretic series of posts and do something a bit more geometric here. I’ll describe the classical Atiyah-Bott fixed point formula for an elliptic complex and one of the applications in the paper. The ultimate goal is for me to understand some of the more recent rigidity results for genera.
1. The Atiyah-Bott fixed point formula
Let be a compact manifold, and suppose given an endomorphism with finitely many fixed points. The classical Lefschetz fixed-point formula counts the number of fixed points via the supertrace of the action of on cohomology . In other words, if is the fixed point set, we have
where is a sign related to the determinant of at .
Using the de Rham isomorphism, the groups are identified with the cohomology of a complex of sections of bundles
This is an example of an elliptic complex of differential operators: in other words, when one takes the symbol sequence at a nonzero cotangent vector, the induced map of vector spaces is exact. It is a consequence of this that the cohomology groups are finite-dimensional.
The Atiyah-Bott fixed point formula is a striking generalization of the previous fact. Consider an elliptic complex of differential operators on ,
where the are vector bundles over . The cohomology groups of this complex are finite-dimensional and provide a generalization (not much of a generalization, actually) of the index of an elliptic operator; they thus often hold significant geometric information about . (more…)
April 9, 2012
1. Vector fields and the Euler characteristic
It is a classical fact that a compact manifold admitting a nowhere vanishing vector field satisfies . One way to prove this is to note that the local flows generated by the vector field are homotopic to the identity, but have no fixed points for small (since the vector field is nonvanishing). By the Lefschetz fixed point theorem, we find that the Lefschetz number of , which is , must vanish.
There is another way of proving this theorem, which uses the theory of elliptic operators instead of the Lefschetz fixed-point theorem. On any -dimensional oriented Riemannian manifold , the Euler characteristic can be computed as the index of the elliptic operator
from even-dimensional differential forms to odd-dimensional ones. Here is exterior differentiation and the formal adjoint, which comes from the metric. One way to see this is to observe that the elliptic operator thus defined is just a “rolled up” version of the usual de Rham complex
In fact, can be defined on the entire space , and there it is self-adjoint (consequently with index zero).
It follows that
The elements in are precisely the harmonic differentials (in fact, is a square root of the Hodge Laplacian ), and by Hodge theory these represent cohomology classes on . It follows thus that
Atiyah’s idea, in his paper “Vector fields on manifolds,” is to use the existence of a nowhere vanishing vector field to get a symmetry of (or a perturbation thereof) to show that its index is zero. (more…)