The fundamental group of generic polynomials (Q1184329)

In the main theorem of this interesting short paper, the authors compute the fundamental group of the space of generic complex polynomials in one variable of degree exactly \((n+1)\). A map \(P: \mathbb{C}\to\mathbb{C}\), given by a polynomial of degree \((n+1)\), is called generic if the derivative map \(P'\) has \(n\) distinct roots and the respective branch points (the image points of the roots) are also distinct. The fundamental group in question is a direct product \(\mathbb{Z}\times H_ n\) and occurs also as a central extension of the integers \(\mathbb{Z}\) by a group \(G_ n\), where \(H_ n\subset G_ n\) are subgroups of the Artin braid group on \(n\) strings.