# A homotopy exact sequence for overconvergent isocrystals

## TL;DR

This paper establishes the exactness of the homotopy sequence for overconvergent p-adic fundamental groups in characteristic p, extending known results from characteristic 0 via a series of reductions.

## Contribution

It proves a new exactness result for overconvergent p-adic fundamental groups in characteristic p, building on analogous results in characteristic 0.

## Key findings

- Exactness of the homotopy sequence for overconvergent p-adic fundamental groups in characteristic p.
- Reduction techniques from characteristic p to characteristic 0 cases.
- Application of rigid analytic methods to overconvergent isocrystals.

## Abstract

In this article we prove exactness of the homotopy sequence of overconvergent $p$-adic fundamental groups for a smooth and projective morphism in characteristic $p$. We do so by first proving a corresponding result for rigid analytic varieties in characteristic $0$, following dos Santos in the algebraic case. In characteristic $p$, we then proceed by a series of reductions to the case of a liftable family of curves, where we can apply the rigid analytic result.