# Directed diagrammatic reducibility

## TL;DR

This paper introduces directed diagrammatic reducibility, a relative concept with significant implications for group theory and topology, extending classical tools and results to this new framework.

## Contribution

It defines directed diagrammatic reducibility, explores its properties, and generalizes key theorems like the Freiheitssatz and Corson-Trace characterization.

## Key findings

- Directed diagrammatic reducibility has strong group theoretic and topological consequences.
- A multi-relator Freiheitssatz is established under this new notion.
- Classical tools are adapted to analyze directed diagrammatic reducibility.

## Abstract

We introduce the notion of directed diagrammatic reducibility which is a relative version of diagrammatic reducibility. Directed diagrammatic reducibility has strong group theoretic and topological consequences. A multi-relator version of the Freiheitssatz in the presence of directed diagrammatic reducibility is given. Results concerning asphericity and $\pi_1$-injectivity of subcomplexes are shown. We generalize the Corson-Trace characterization of diagrammatic reducibility to directed diagrammatic reducibility. We compare diagrammatic reducibility of relative presentations to directed diagrammatic reducibility. Classical tools for showing diagrammatic reducibility, such as the weight test, the max/min test, and small cancellation techniques are adapted to directed diagrammatic reducibility. The paper ends with some applications to labeled oriented trees.