On the monodromies of a polynomial map from C2 to C

Let f : C2→C be a polynomial function. It is well known that there exists a finite set A⊂C such that the restriction of f to C2−f−1(A) is a differentiable fibration onto C−A. Following Broughton in (Proc. Symp. Pure Math. 40 (1983) 167) we call the smallest of such Aʹs the set of atypical values of f and write it Af. Let F be a generic fiber of f. The main goal of this article is to describe the monodromy on H1(F,Z) around an atypical value a∈Af. For that purpose we define and study a monodromic filtration on the homology with coefficients in Z : 0⊂M−1⊂M0⊂M1⊂M2=H1(F,Z). The term M−1 is added to allow for the boundary of F. We introduce a compact model L̂a for the smooth part of the reduced curve associated to the affine fiber f−1(a). One important result of this article is theorem (8.12) which shows how H1(L̂a,Z) gives (via the transfer homomorphism) a precise description of the invariant cycles in H1(F,Z).