Horizontal decompositions, II

We complete the classification of the smooth, closed, oriented 4-manifolds having Euler characteristic less than four and a horizontal handlebody decomposition of genus one.We use the classification result to find a large family of rational homology ball smoothings of cyclic quotient singularities which can be smoothly embedded into the complex projective plane.Our family contains all such rational balls previously known to embed into CP 2 and infinitely many more.We also show that a rational ball of our family admits an almost-complex embedding in CP 2 if and only if it admits a symplectic embedding. 57K40; 57K43, 57R17We introduce some notation in order to give a more concise statement of Theorem 1.2.When we write B p;q we shall always assume that p and q are coprime integers, but not necessarily satisfying p > q 0.