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 {M} be a compact manifold, and suppose given an endomorphism {f: M \rightarrow M} with finitely many fixed points. The classical Lefschetz fixed-point formula counts the number of fixed points via the supertrace of the action of {f} on cohomology {H^*(M; \mathbb{R})}. In other words, if {F} is the fixed point set, we have

\displaystyle \mathrm{Tr} (f)|_{H^*(M; \mathbb{R})} = \sum_{p \in F} (-1)^{\sigma(p)},

where {\sigma} is a sign related to the determinant of {1 - df} at {p}.

Using the de Rham isomorphism, the groups {H^*(M; \mathbb{R})} are identified with the cohomology of a complex of sections of bundles

\displaystyle 0 \rightarrow \Gamma(1) \stackrel{d}{\rightarrow} \Gamma(T^* M) \stackrel{d}{\rightarrow} \dots.

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 {M},

\displaystyle 0 \rightarrow \Gamma(E_0 ) \rightarrow \Gamma(E_1) \rightarrow \dots \rightarrow \Gamma(E_r) \rightarrow 0,

where the {E_i} are vector bundles over {M}. 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 {M}. (more…)

This is the second post devoted to describing some of the ideas in Atiyah’s paper “Vector fields on manifolds.” Last time, we saw that one could prove the classical vanishing of the Euler characteristic on a manifold admitting a nowhere zero vector field using the symmetries of the de Rham complex. In this post, I’ll describe how analogous methods lead to some of the deeper results in the paper.

1. The case of a field of planes

One of the benefits of Atiyah’s idea of using symmetries of differential operators is that it gives us a host of other results, which are not connected with the Lefschetz fixed-point theorem.

For instance:

Theorem 3 Let {M} be a compact manifold admitting an oriented two-dimensional subbundle {F \subset TM}. Then {\chi(M)} is even.

The proof of this result starts off as before. Yesterday, we observed that the Euler characteristic of a Riemannian manifold {M} can be obtained as the index of the elliptic operator

\displaystyle D = d + d^* : \Omega^{even}(M) \rightarrow \Omega^{odd}(M).

The operator {D} (obtained by “rolling up” the de Rham complex, whose index is precisely {\chi(M)}) is a map of the global sections {\bigwedge^{even} T^*M \rightarrow \bigwedge^{odd} T^* M}. As we saw yesterday, the symbol of this operator {D} is precisely given by left Clifford multiplication. In other words, the symbol of {D} at a cotangent vector {v \in T_x^* M} is precisely given by left Clifford multiplication {L_v} by {v} on {\bigwedge^{even} T^*_x M = \mathrm{Cl}^0(T^*_x M)}.

The method Atiyah uses to construct symmetries of {D} is to use the simple observation that left and right Clifford multiplication commute. This enables him to construct an operator {I} commuting with {D} such that {I^2 = - 1}, thus—approximately—endowing the kernel and cokernel of {D} with a complex structure. (more…)

1. Vector fields and the Euler characteristic

It is a classical fact that a compact manifold {M} admitting a nowhere vanishing vector field satisfies {\chi(M) = 0}. One way to prove this is to note that the local flows {\phi_\epsilon} generated by the vector field are homotopic to the identity, but have no fixed points for {\epsilon } small (since the vector field is nonvanishing). By the Lefschetz fixed point theorem, we find that the Lefschetz number of {\phi_\epsilon}, which is {\chi(M)}, 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 {n}-dimensional oriented Riemannian manifold {M}, the Euler characteristic can be computed as the index of the elliptic operator

\displaystyle D = d + d^* : \Omega^{even}(M) \rightarrow \Omega^{odd}(M)

from even-dimensional differential forms to odd-dimensional ones. Here {d} is exterior differentiation and {d^*} 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

\displaystyle 0 \rightarrow \Omega^0(M) \rightarrow \Omega^1(M) \rightarrow \dots.

In fact, {d + d^*} can be defined on the entire space {\Omega^\bullet(M)}, and there it is self-adjoint (consequently with index zero).

It follows that

\displaystyle \mathrm{index}D = \dim \ker D - \dim \mathrm{coker }D = \dim \ker (d + d^*)|_{\Omega^{even}(M)}- \dim \ker (d + d^*)|_{\Omega^{odd}(M)}.

The elements in {\ker d + d^*} are precisely the harmonic differentials (in fact, {d + d^*} is a square root of the Hodge Laplacian {dd^* + d^* d}), and by Hodge theory these represent cohomology classes on {M}. It follows thus that

\displaystyle \mathrm{index} D = \dim H^{even}(M) - \dim H^{odd}(M).

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 {D} (or a perturbation thereof) to show that its index is zero. (more…)

The next application I want to talk about here of Fourier analysis is to (a basic case of) ellipic regularity. Later we will use refinements of these techniques to obtain all kinds of estimates. Anyway, for now, a partial differential operator

\displaystyle P = \sum_{a: |a| \leq k} C_a D^a

is called elliptic if the homogeneous polynomial

\displaystyle \sum_{a: |a| = k} C_a \xi^a, \quad \xi = (\xi_1, \dots, \xi_n)

has no zeros outside the origin. For instance, the Laplace operator is elliptic. Later I will discuss how this generalizes to other PDEs, and how this polynomial becomes the symbol of the operator. For the moment, though let’s define {Q(\xi) = \sum_{a: |a| \leq k} C_a (2 \pi i \xi)^a}. The definition of {Q} such that

\displaystyle \widehat{ Pf } = Q \hat{f},

and we know that {|Q(\xi)| \geq \epsilon |\xi|^k} for {|\xi|} large enough. This is a very important fact, because it shows that the Fourier transform of {Pf} exerts control on that of {f}. However, we cannot quite solve for {\hat{f}} by dividing {\widehat{Pf}} by {Q} because {Q} is going to have zeros. So define a smoothing function {\varphi} which vanishes outside a large disk {D_r(0)}. Outside this disk, an estimate {|Q(\xi)| \geq \epsilon |\xi|^k} will be assumed to hold. (more…)