The mapping class group and the Meyer function for plane curves

TL;DR

This paper introduces a new mapping class group for plane curves of degree d, establishes the existence of a unique Meyer function on this group, and applies it to define local signatures for certain 4D fiber spaces.

Contribution

It defines the mapping class group for plane curves of degree d and proves the existence and uniqueness of the Meyer function on this group, with applications to fiber space signatures.

Findings

Existence and uniqueness of the Meyer function for the defined group

Application of the Meyer function to local signatures in 4D fiber spaces

Computations of local signatures for specific cases

Abstract

For each d>=2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d=4, using our Meyer function, we can define the local signature for 4-dimensional fiber spaces whose general fibers are non-hyperelliptic compact Riemann surfaces of genus 3. Some computations of our local signature will be given.