# Minimal genus problem for $T^{2}$-bundles over surfaces

## TL;DR

This paper determines the minimal genus of embedded surfaces in certain 4-manifolds, showing the adjunction inequality is not always sharp and providing exact genus values for all classes.

## Contribution

It completely computes the minimal genus function for $ ext{surface} imes T^2$ and demonstrates cases where the adjunction inequality is not sharp.

## Key findings

- Exact minimal genus values are obtained for all classes in $ ext{surface} imes T^2$.
- The adjunction inequality does not always give sharp bounds.
- Constructed embedded surfaces match the computed minimal genus.

## Abstract

For any positive integer $g$, we completely determine the minimal genus function for $\Sigma_{g}\times T^{2}$. We show that the lower bound given by the adjunction inequality is not sharp for some class in $H_{2}(\Sigma_{g}\times T^{2})$. However, we construct a suitable embedded surface for each class and we have exact values of minimal genus functions.