  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.
