# Minimal genus four manifolds

## TL;DR

This paper demonstrates that for any given group, there exists a 4-manifold with that fundamental group whose trisection genus attains the lower bound established by Chu and Tillmann, advancing understanding of 4-manifold topology.

## Contribution

It constructs 4-manifolds with prescribed fundamental groups that achieve the minimal possible trisection genus according to Chu and Tillmann's bound.

## Key findings

- Existence of 4-manifolds with any fundamental group and minimal trisection genus
- Validation of Chu-Tillmann's lower bound as sharp
- Contribution to classification of 4-manifolds by genus and fundamental group

## Abstract

In 2018, M. Chu and S. Tillmann gave a lower bound for the trisection genus of a closed 4-manifold in terms of the Euler characteristic of $M$ and the rank of its fundamental group. We show that given a group $G$, there exist a 4-manifold $M$ with fundamental group $G$ with trisection genus achieving Chu-Tillmann's lower bound.