When studying the Atiyah-Singer index theorem one usually sees four main examples.
- Atiyah-Singer operator: Its topological index is the \(\hat{A}\)-genus and its analytical index can be related to scalar curvature. This shows that the \(\hat{A}\)-genus of a manifold is an obstruction to the existence of a Riemannian metric of positive scalar curvature on it.
- Signature operator: Its topological index is the \(L\)-genus and its analytical index is the signature of the manifold. This recovers Hirzebruchs signature theorem as a special case of the Atiyah-Singer index theorem.
- Dolbeault operator: Its analytical index is the holomorphic Euler characteristic and comparison with the topological index recovers the Hirzebruch-Riemann-Roch theorem.
- Euler characteristic operator: Its analytical index is the Euler characteristic and comparison with the topological index recovers in dimension 2 the Gauss-Bonnet theorem.
Now I personally enjoy the K-homological approach to the Atiyah-Singer index theorem a lot. The main feature of this approach is that elliptic differential operators, i.e. the ones for which Atiyah-Singer applies, define classes in it. So each of the above four operators defines a class in a suitable K-homology group, and one can now investigate this class.
- In the first three cases it turns out that the class is highly non-trivial. In fact, one can show that each of them is an orientation class (in a suitably interpreted sense). For example, for the Atiyah-Singer operator this implies that cap product by it is an isomorphism \(K^*(M) \to K_{m-*}(M)\).
- But in the fourth case, i.e. for the Euler characteristic operator, it is exactly the other way round: It is as trivial as it can be! This was proven by Jonathan Rosenberg (arXiv:math/9806073) and the concrete statement is that the K-homological class of the Euler characteristic operator is in the image of the map \(K_0(\mathrm{pt}) \to K_0(M)\) induced by the inclusion of a point.