Three-dimensional manifolds which are the boundary of a normal singularity \(z^k- f(x,y)\)
The purpose of this paper is to describe the connections between the topology of the boundary \({L_f}^k\) of the analytic germ \({z^k}-f(x,y)\) and the topology-type of \(f\). The author's method consists of equipping \({L_f}^k\) with the Waldhausen decomposition and with the open-book decomposition induced respectively by Waldhausen decomposition of \(S_{\varepsilon}^3 \backslash K_f\) and by the Milnor fibration of \(f\), via the cover \(R\). The author computes explicitly the corresponding Waldhausen graph of \({L_f}^k\) by using the quasi-periodic monodromy of the Milnor fibration and an invariant of the conjugacy class of quasi-periodic diffeomorphism. The author shows the following ``finiteness theorem''. Finite theorem: Let \(M\) be a 3-manifold and let \(k\) be any integer \({\geq}\) 3. There exists, up to topological-type, a finite number (perhaps equal to zero) of reduced analytic germs \(f: C^2 \rightarrow C, 0\) such that \(M\) is homeomorphic to \(L_f^k\). Furthermore, there exists an algorithm to draw up the list of the topological-type of such germs \(f\).
scientific article; zbMATH DE number 1336868