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…)