# A relative version of Gieseker's problem on stratifications in characteristic $p>0$

## TL;DR

This paper investigates how the vanishing of a functoriality morphism for the étale fundamental group in characteristic p>0 influences the fundamental groups of stratifications, revealing new connections in algebraic geometry.

## Contribution

It establishes a link between the vanishing of functoriality morphisms and the fundamental groups of stratifications in characteristic p>0, extending Gieseker's problem.

## Key findings

- Vanishing of the functoriality morphism implies the same for stratification fundamental groups.
- Provides new insights into the structure of fundamental groups in positive characteristic.
- Extends Gieseker's problem to a relative setting in algebraic geometry.

## Abstract

We prove that the vanishing of the functoriality morphism for the \'etale fundamental group between smooth projective varieties over an algebraically closed field of characteristic $p>0$ forces the same property for the fundamental groups of stratifications. 2nd version: Acknowledgement added.