Topological index

Remarks

In cohomology, the topological index is the integral over $T^*M$ of $\operatorname{ch}[\sigma(P)] \cdot \pi^* \operatorname{Td}(TM \otimes \C)$, up to a sign fixed by orientation conventions. Here $\operatorname{ch}$ is the Chern character and $\operatorname{Td}$ is the Todd class. Let $M$ be a closed oriented Riemannian manifold of even dimension and $P = d + d^*$ from even to odd forms. Then the topological index is the Euler number $\langle e(TM), [M] \rangle$, which is the Euler characteristic of $M$.

Example: let $P \colon C^\infty(M; E) \to C^\infty(M; F)$ be the operator of order $0$ given by a bundle isomorphism $\phi \colon E \to F$. Its symbol is $\pi^*\phi$, which is invertible on all of $T^*M$, so $[\sigma(P)] = 0$ in $K(T^*M)$ and $\operatorname{t-ind} P = 0$. This agrees with the analytic index, since $P$ is bijective.

Non-example: for a non-elliptic operator, the principal symbol is not invertible at some nonzero covector. Then $\sigma(P)$ does not define a class in $K(T^*M)$, and there is no topological index.