# Stable existence of incompressible 3-manifolds in 4-manifolds

## TL;DR

This paper establishes conditions under which incompressible 3-manifolds can stably embed in 4-manifolds, providing algebraic criteria and a generalization of the Lickorish--Wallace theorem for covers.

## Contribution

It introduces an algebraic-topological splitting criterion for stable embeddings of 3-manifolds in 4-manifolds and generalizes the Lickorish--Wallace theorem to regular covers.

## Key findings

- Provides a splitting criterion based on orientation classes and universal covers.
- Generalizes the Lickorish--Wallace theorem to equivariant settings.
- Establishes conditions for stable existence of incompressible 3-manifolds in 4-manifolds.

## Abstract

Given an injective amalgam at the level of fundamental groups and a specific 3-manifold, is there a corresponding geometric-topological decomposition of a given 4-manifold, in a stable sense? We find an algebraic-topological splitting criterion in terms of the orientation classes and universal covers. Also, we equivariantly generalize the Lickorish--Wallace theorem to regular covers.