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…)