The Milnor number is a powerful invariant of an isolated, complex, affine hypersurface singularity. It provides data about the local, ambient, topological-type of the hypersurface, and the constancy of the Milnor number throughout a family implies that Thom's \(a_f\) condition holds and that the local, ambient, topological-type is constant in the family. Much of the usefulness of the Milnor number is due to the fact that it can be effectively calculated in an algebraic manner.

The Lê cycles and numbers are a generalization of the Milnor number to the setting of complex, affine hypersurface singularities, where the singular set is allowed to be of arbitrary dimension. As with the Milnor number, the Lê numbers provide data about the local, ambient, topological-type of the hypersurface, and the constancy of the Lê numbers throughout a family implies that Thom's \(a_f\) condition holds and that the Milnor fibrations are constant throughout the family. Again, much of the usefulness of the Lê numbers is due to the fact that they can be effectively calculated in an algebraic manner.

In this work, we generalize the Lê cycles and numbers to the case of hypersurfaces inside arbitrary analytic spaces. We define the Lê-Vogel cycles and numbers, and prove that the Lê-Vogel numbers control Thom's \(a_f\) condition. We also prove a relationship between the Euler characteristic of the Milnor fibre and the Lê-Vogel numbers. Moreover, we give examples which show that the Lê-Vogel numbers are effectively calculable.

In order to define the Lê-Vogel cycles and numbers, we require, and include, a great deal of background material on Vogel cycles, analytic intersection theory, and the derived category. Also, to serve as a model case for the Lê-Vogel cycles, we recall our earlier work on the Lê cycles of an affine hypersurface singularity.