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 3Let be a compact manifold admitting an oriented two-dimensional subbundle . Then is even.

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

The operator (obtained by “rolling up” the de Rham complex, whose index is precisely ) is a map of the global sections . As we saw yesterday, the symbol of this operator is precisely given by left Clifford multiplication. In other words, the symbol of at a cotangent vector is precisely given by left Clifford multiplication by on .

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