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))*.

Article

Milnor Numbers and the Topology of Polynomial Hypersurfaces

Faculty Publications - Mathematics
Document Type

Article
Publication Date

1-1-1988
Disciplines

Abstract

External Access URL

http://link.springer.com/article/10.1007/BF01404452
Citation Information

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