Braid Group, Algebraic Surfaces and Fundamental Groups of complements of Branch Curves

Abstract

This paper will appear in the Santa Cruz proceedings. An overview of the braid group techniques in the theory of algebraic surfaces, from Zariski to the latest results, is presented. An outline of the Van Kampen algorithm for computing fundamental groups of complements of curves and the modification of Moishezon-Teicher regarding branch curves of generic projections are given. The paper also contains a description of a quotient of the braid group, namely $\tilde B_n$ which plays an important role in the description of fundamental groups of complements of branch curves.It turns out that all such groups are ``almost solvable'' $\tildeB_n$-groups. Finally, possible applications to the study of moduli spaces of surfaces of general type are described and new examples of positive signature spin surfaces whose fundamental groups can be computed using the above algorithm (Galois cover of Hirzebruch surfaces) are presented.