Let F: ℂn + 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 genericc∈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)).
Milnor Numbers and the Topology of Polynomial HypersurfacesFaculty Publications - Mathematics
External Access URLhttp://link.springer.com/article/10.1007/BF01404452
Citation InformationBroughton, S.A. (1988). Milnor numbers and the topology of polynomial hyper surfaces. In Inventiones Mathematicae, 92(2), 217-241.