Rational homology disk smoothings and Lefschetz fibrations

TL;DR

This paper extends the theory of Lefschetz fibrations by incorporating genus into fibers, providing new relations in mapping class groups that correspond to rational homology disk smoothings of specific surface singularities.

Contribution

It introduces a genus to the fibers of Lefschetz fibrations and establishes new relations in mapping class groups for rational homology disk smoothings.

Findings

Derived new relations in genus-1 surface mapping class groups

Connected Lefschetz fibrations to rational homology disk smoothings

Extended previous results to more general fiber structures

Abstract

In this article, we generalize the results discussed in [arXiv:1004.3762] by introducing a genus to generic fibers of Lefschetz fibrations. That is, we give families of relations in the mapping class groups of genus-1 surfaces with boundaries that represent rational homology disk smoothings of weighted homogeneous surface singularities whose resolution graphs are $3$-legged with a bad central vertex.