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…)