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 Thorn's a/ con-
dition 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 Le 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 Le numbers provide data about the
local, ambient, topological-type of the hypersurface, and the constancy of the Le numbers
throughout a family implies that Thorn's a/ condition holds and that the Milnor fibrations
are constant throughout the family. Again, much of the usefulness of the Le numbers is due
to the fact that they can be effectively calculated in an algebraic manner.
In this work, we generalize the Le cycles and numbers to the case of hypersurfaces inside
arbitrary analytic spaces. We define the Le-Vogel cycles and numbers, and prove that the Le-
Vogel numbers control Thorn's a/ condition. We also prove a relationship between the Euler
characteristic of the Milnor fibre and the Le-Vogel numbers. Moreover, we give examples
which show that the Le-Vogel numbers are effectively calculable.
In order to define the Le-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 Le-Vogel cycles, we recall our earlier work
on the Le cycles of an affine hypersurface singularity.
1991 Mathematics Subject Classification. 32B15, 32C35, 32C18, 32B10
Key words and phrases. Gap sheaf, Vogel cycle, Milnor fibre and number, Le cycles and
numbers, vanishing cycles, perverse sheaves, Thorn's af condition, Le-Vogel cycles and numbers
Received by the editor May 1, 2001