Regularity at infinity of real mappings and a Morse-Sard theorem

We prove a new Morse-Sard type theorem for the asymptotic critical values of semi-algebraic mappings and a new fibration theorem at infinity for $C^2$ mappings. We show the equivalence of three different types of regularity conditions which have been used in the literature in order to control the asymptotic behaviour of mappings. The central role of our picture is played by the $t$-regularity and its bridge toward the $\rho$-regularity which implies topological triviality at infinity

Vanishing cycles and singularities of meromorphic functions

We study vanishing cycles of meromorphic functions This gives a new and unitary point of view extending the study of the topology of holomorphic germs as initiated by Milnor in the sixties and of the global topology of polynomial functions which has been advanced more recently We dene singularities along the poles with respect to a certain weak stratication and prove local and global bouquet structure in case of isolated singularities In general splitting of vanishing homology at singular points and global PicardLefschetz phenomena occu

Singularities at infinity and their vanishing cycles, II : monodromy

Let f C n C be any polynomial function By using global polar methods we introduce models for the bers of f and we study the monodromy at atypical values of f including the value innity We construct a geometric monodromy with controlled behavior and dene global relative monodromy with respect to a general linear form We prove localization results for the relative monodromy and derive a zetafunction formula for the monodromy around an atypical value We compute the relative zeta function in several cases and emphasize the dierences to the classical local situatio

On the geometry of regular maps from a quasi-projective surface to a curve

We explore consequences of the triviality of the monodromy group, using the condition of purity of the mixed Hodge structure on the cohomology of the surface X

Betti bounds of polynomials

We initiate a classification of polynomials f : Cn - C of degree d having the top Betti number of the general fibre close to the maximum. We find a range in which the polynomial must have isolated singularities and another range where it may have at most one line singularity of Morse transversal type. Our method uses deformations into particular pencils with non-isolated singularitie