Homotopy type of complements of fiber-type curves

TL;DR

This paper investigates the homotopy types of complements of fiber-type plane curves, providing a detailed description under certain conditions and addressing a question about whether the fundamental group and Euler characteristic determine the homotopy type.

Contribution

It offers a full description of the homotopy type of fiber-type curve complements and partially answers a question about the determining factors of their homotopy type.

Findings

Provided a full description of homotopy types under restrictions

Partially answered Libgober's question on homotopy type determination

Showed homotopy type is not solely determined by fundamental group and Euler characteristic

Abstract

In this note we study the homotopy type of the complement of a plane projective curve of fiber-type. Roughly speaking, a curve of fiber-type is a finite union of fibers of a pencil. Under some restrictions, a full description of their homotopy type is given. In the context of fiber-type curves, we also partially answer in the positive a question originally posed by Libgober on the homotopy type of complements of plane curves, namely, whether or not the homotopy type of a curve complement is determined by the fundamental group and its Euler characteristic.