# Profinite detection of 3-manifold decompositions

## TL;DR

This paper demonstrates that the profinite completion of a 3-manifold's fundamental group uniquely encodes its prime and JSJ decompositions, revealing deep connections between algebraic and topological structures.

## Contribution

It establishes that the profinite completion determines the prime and JSJ decompositions of closed, orientable 3-manifolds, advancing understanding of their algebraic-topological correspondence.

## Key findings

- Profinite completion determines the Kneser--Milnor decomposition.
- For irreducible manifolds, it determines the JSJ decomposition.
- Provides a new algebraic method to analyze 3-manifold topology.

## Abstract

The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.