# Ribbon homology cobordisms

## TL;DR

This paper investigates 4-dimensional homology cobordisms without 3-handles, exploring their interactions with geometric and topological invariants, and deriving obstructions and applications related to knot theory and Dehn surgeries.

## Contribution

It introduces new obstructions to homology cobordisms without 3-handles and generalizes results on knot Floer homology under ribbon concordances.

## Key findings

- Obstructions derived from Thurston geometries and Floer homologies.
- Generalization of knot Floer homology behavior under ribbon concordances.
- Applications to Dehn surgery problems.

## Abstract

We study 4-dimensional homology cobordisms without 3-handles, showing that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. Using these, we derive obstructions to such cobordisms. As one example of these obstructions, we generalize other recent results on the behavior of knot Floer homology under ribbon concordances. Finally, we provide topological applications, including to Dehn surgery problems.