Signature operator

Summary

The signature operator of a closed oriented Riemannian manifold of dimension $4k$ is the operator $d + d^*$ on complex differential forms. It is restricted to the forms that are even under a natural involution built from the Hodge star. It is an elliptic operator, and its index equals the signature of the manifold, the signature of the intersection form on middle cohomology. Applied to it, the Atiyah–Singer index theorem gives the Hirzebruch signature theorem. It is one of the classical operators of index theory, together with the de Rham, Dolbeault and spin Dirac operators.