Horizontal decompositions and smooth embeddings of rational homology balls

We start by introducing the problem of constructing smooth embeddings in CP^2 of some rational homology balls (abbr. rational balls) bounded by lens spaces. After mentioning the list of embeddings that were constructed in earlier works by several other authors, we show how to extend such list: namely, we use Donaldson's theorem on the intersection forms of smooth 4-manifolds to get an obstruction for most of the rational balls taken into consideration, and on the other hand we exhibit smooth embeddings for some unobstructed balls. The latter construction makes use of a special kind of handle decompositions: in the second part, we define them as horizontal decompositions and show their main properties; in particular, we show that every 4-dimensional cobordism admits a horizontal decomposition. Finally, we classify (in some simple cases) the 4-manifolds that admit a horizontal decomposition of a given type, and after observing how such decompositions induce smooth embeddings of rational balls in the resulting 4-manifolds, we compute the full list of embeddings in CP^2 that arise in this way. This procedure leads to a further extension of the list of known embeddings.