#### Title

Milnor Numbers and the Topology of Polynomial Hypersurfaces

#### Document Type

Article

#### Publication Date

1-1-1988

#### Abstract

Let *F*: ℂ^{n + 1}→ℂ be a polynomial. The problem of determining the homology groups *H* _{ q } *(F* ^{ −1 } *(c)), c ∈*ℂ, in terms of the critical points of* F* is considered. In the “best case” it is shown, for a certain generic class of polynomials (tame polynomials), that for all* c∈*ℂ,*F* ^{ −1 } *(c)* has the homotopy type of a bouquet of μ-μ^{ c } *n*-spheres. Here μ is the sum of all the Milnor numbers of* F* at critical points of* F* and μ^{ c } is the corresponding sum for critical points lying on *F* ^{ −1 } *(c)*. A “second best” case is also discussed and the homology groups* H* _{ q } *(F* ^{ −1 } *(c))* are calculated for generic*c∈*ℂ. This case gives an example in which the critical points “at infinity” of* F* must be considered in order to determine the homology groups *H* _{ q } *(F* ^{ −1 } *(c))*.

#### Recommended Citation

Broughton, S.A. (1988). Milnor numbers and the topology of polynomial hyper surfaces. In Inventiones Mathematicae, 92(2), 217-241.