Index of a singular point of a vector field or of a 1-form on an orbifold

Abstract

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related to the Euler characteristic through the classical Poincaré–Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related to corresponding analogs of indices of singular points. Earlier, a notion of the universal Euler characteristic of an orbifold was defined. It takes values in a ring R \mathcal {R} , as an Abelian group freely generated by the generators, corresponding to the isomorphism classes of finite groups. Here the universal index of an isolated singular point of a vector field or of a 1-form on an orbifold is defined as an element of the ring R \mathcal {R} . For this index, an analog of the Poincaré–Hopf theorem holds.