Summary
LetF: ℂn + 1→ℂ be a polynomial. The problem of determining the homology groupsH q (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-μc n-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 groupsH q (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 groupsH q (F −1 (c)).
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Broughton, S.A. Milnor numbers and the topology of polynomial hypersurfaces. Invent Math 92, 217–241 (1988). https://doi.org/10.1007/BF01404452
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DOI: https://doi.org/10.1007/BF01404452