Topological Monodromy

Abstract

A triple (M,D,ψ) is called a degenerating family of Riemann surfaces of genus g (abbreviated as degenerating family of genus g) if M is a complex surface, \( D=\{\xi \in \mathbf{C}||\xi|<1\},\)\(\psi:M\rightarrow D\) is a surjective proper holomorphic map, for each \(\psi:M\rightarrow D \), the fiber \( {F _\xi}={\psi^{-1}\Psi}(\xi)\) is connected, and \(\psi{|_{M^*}}:{M^*}\rightarrow {D^*}\) is a smooth (i.e.C^∞) fiber bundle whose fiber is a Riemann surface of genus g, where \({D^*}=D-\{0\}\Psi\) and \({M^*}-{\psi^{-1}}(0)\).