Symplectic Lefschetz fibrations with arbitrary fundamental groups. With an appendix by Ivan Smith (Q1609776)

The authors give an explicit construction of a symplectic Lefschetz fibration (SLF) whose total space is a smooth compact \(4\)-manifold with a prescribed fundamental group, following the existence result by \textit{S. Donaldson} [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 309-314 (1998; Zbl 0909.53018)] and the result on the realization of any finitely presentable group as a fundamental group by \textit{R. Gompf} [Ann. Math., II. Ser. 142, 527-595 (1995; Zbl 0849.53027)]. The construction in the present article is based on an algebraic geometric method for creating positive relations among right-handed Dehn twists. The method is provided by a careful study of structures of the mapping class group \(M^n_{g,r}\) of a smooth genus \(g\) surface with \(n\) punctures and \(r\) boundary components, and symplectic connected sums for symplectic \(4\)-manifolds. Proofs of the main theorems are purely topological, based on exploiting the correspondence between subgroups of \(M^n_g(=M^n_{g,0})\) and graphs of vanishing cycles, and are achieved by showing comprehensible illustrations. In the last section, the authors prove that the mapping class group \(M^1_1\) is isomorphic to \(\widetilde{SL}(2, \mathbb Z)\). This fact is used to study numerical properties of Dehn twist relations in elliptic SLFs and the authors prospect further study of \(M^1_g\) for SLFs with fibers of higher genus, besides I. Smith's result on the non-existence of SLFs with monodromy contained in the Torelli group.