# $2$-stratifolds with fundamental group $\mathbb{Z}$

## TL;DR

This paper classifies trivalent 2-stratifolds with fundamental group Z, providing a clear characterization and an efficient algorithm to determine this property from their associated labelled graphs.

## Contribution

It offers a classification of trivalent 2-stratifolds with fundamental group Z and introduces an algorithm for identifying this from their labelled graphs.

## Key findings

- Classification of trivalent 2-stratifolds with fundamental group Z
- Development of an efficient graph-based decision algorithm
- Extension of previous 1-connected case results

## Abstract

$2$-stratifolds are a generalization of $2$-manifolds that occur as objects in applications such as in TDA. These spaces can be described by an associated bicoloured labelled graph. In previous papers we obtained a classification of 1-connected trivalent $2$-stratifolds. In this paper we classify trivalent $2$-stratifolds that have fundamental group $\mathbb{Z}$. This classification implicitly gives an efficient algorithm that applies to a bicoloured labelled graph to decide wether the associated $2$-stratifold has fundamental group $\mathbb{Z}$.