Fundamental groups of links of isolated singularities (Q2922788)

In this interesting paper the authors study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities.NEWLINENEWLINEThe main result is the following:NEWLINENEWLINETheorem. For every finitely presented group \(G\) there is an isolated, \(3\)-dimensional, complex singularity (\(0\in X_{G}\)) with link \(L_{G}\) such that \(\pi _{1}(L_{G})\cong G\).NEWLINENEWLINEThe authors consider that this result is a strong exception to the principle formulated by \textit{M. Goresky} and \textit{R. MacPherson} [Stratified Morse theory. Berlin etc.: Springer-Verlag (1988; Zbl 0639.14012)]:NEWLINENEWLINE``Philosophically, any statement about the projective variety or its embedding really comes from a statement about the singularity at the point of the cone. Theorems about projective varieties should be consequences of more general theorems about singularities which are no longer required to be conical. ''NEWLINENEWLINEIn order to prove the above theorem, the authors show that for every finitely presented group \(G\) there is a complex projective surface \(S\) with simple normal crossing singularities only, so that the fundamental groups of \(S\) is isomorphic to \(G\).NEWLINENEWLINEAt the end of the paper the authors prove that a finitely presented group \(G\) is \(\mathbb{Q}\)-superperfect (has vanishing rational homology in dimensions \(1\) and \(2\)) if and only if \(G\) is isomorphic to the fundamental group of the link of a rational \(6\)-dimensional complex singularity.