Milnor numbers and the topology of polynomial hypersurfaces

SummaryLetF: ℂn + 1→ℂ be a polynomial. The problem of determining the homology groupsHq(F−1(c)), c ∈ℂ, in terms of the critical points ofF is considered. In the “best case” it is shown, for a certain generic class of polynomials (tame polynomials), that for allc∈ℂ,F−1(c) has the homotopy type of a bouquet of μ-μcn-spheres. Here μ is the sum of all the Milnor numbers ofF at critical points ofF and μc is the corresponding sum for critical points lying onF−1(c). A “second best” case is also discussed and the homology groupsHq(F−1(c)) are calculated for genericc∈ℂ. This case gives an example in which the critical points “at infinity” ofF must be considered in order to determine the homology groupsHq(F−1(c)).