# Fundamental groups and path lifting for algebraic varieties

## TL;DR

This paper investigates fundamental groups of algebraic varieties, focusing on how surjectivity on π₁ behaves under base change, the relationship between Zariski and Euclidean topologies, and the conditions for path lifting in morphisms.

## Contribution

It provides new insights into the behavior of fundamental groups under base change, compares topological notions, and characterizes morphisms with path lifting properties.

## Key findings

- Surjectivity on π₁ is preserved under certain base changes.
- Openness in Zariski topology relates to Euclidean topology.
- Criteria for morphisms to have the path lifting property.

## Abstract

We study 3 basic questions about fundamental groups of algebraic varieties. For a morphism, is being surjective on $\pi_1$ preserved by base change? What is the connection between openness in the Zariski and in the Euclidean topologies? Which morphisms have the path lifting property?