Irregular fibers of complex polynomials in two variables (Q1884575)

Let \(f:\mathbb C^2 \rightarrow \mathbb C\) be a polynomial and \(\mathcal B \subset \mathbb C\) the bifurcation set. Let \(F_c=f^{-1}(c)\). The fiber \(F_c\) is called irregular if \(c \in \mathcal B\). Let \(D_{\varepsilon}(c) \subset \mathbb C\) be the disc of radius \(\varepsilon\) centered at \(c\) and \(T_c=f^{-1}(D_{\varepsilon}(c))\). The value \(c \in \mathbb C\) is called regular at infinity if there exist a compact set \(K\) in \(\mathbb C^2\) such that the restriction of \(f:T_c \setminus K \rightarrow D_{\varepsilon}(c)\) is locally trivial. The morphism \(j_c:H_1(F_c) \rightarrow H_1(T_c)\) induced by the inclusion is studied. It is proved that it is an isomorphism if and only if \(c\) is regular at infinity. Necessary and sufficient conditions for this morphism to be injective respectively surjective are given. The results are applied to the study of vanishing and invariant cycles.