Geometric bases and topological equivalence

There are many algebraic and topological invariants associated to a singular point of a complex analytic function. The intent here is to discuss some of these invariants and the topological classification of singularities. Specifically, we establish that the topological type is determined by the Lefschetz vanishing cycles obtained by unfolding the singularity and certain local monodromy operators defined by Gabrielov. In Brieskorn's terminology singularities with the same geometric bases are topologically indistinguishable. Thus the higher invariants in the hierarchy of Brieskorn are necessary to understand the geometry of higher singularities. As a corollary to our main theorem, we obtain the result of Lê-Ramanujam which states that the topological type is constant in a oneparameter family of singularities with constant Milnor number.