Horizontal decompositions, II

Abstract

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. INTRODUCTIONLet B p,q , where p > q ≥ 0 are coprime integers, be the rational homology ball smoothing of the cyclic quotient singularity 1 p 2 (pq − 1, 1). The rational balls B p,q were used in the smooth rational blow-down construction [4] and in its symplectic counterpart [16]. They played an important role in the construction of exotic 4-manifolds, starting with the papers [14,3]. The problem of embedding the B p,q 's, smoothly or symplectically in a 4-manifold was considered in [7,9,8]. As pointed out in [2, Section 2], the existence of holomorphic embeddings B p,q → CP 2 for certain (p, q)'s follows from results in algebraic geometry [6], and explicit symplectic embeddings can be constructed. Moreover, Evans and Smith [2] showed that no further symplectic embeddings exist beyond those which are already known. Owens [13] showed that infinitely many B p,q 's admit smooth embeddings in CP 2 but by [2] they cannot be symplectically embedded. The authors extended Owens' family [11] and proved the non-existence of almost complex embeddings [10] without relying on [2]. The present paper is a natural continuation of [12], where we introduced certain handlebody decompositions that we call horizontal, classified the closed 4-manifolds with the simplest horizontal decompositions and in doing so we recovered infinitely many of the known smooth embeddings B p,q → CP 2 .This paper consists of two parts. In the first part we complete the classification of the smooth, closed, oriented 4-manifolds X with Euler characteristic χ(X ) ≤ 3 and a horizontal decomposition of genus 1. In the second part we exploit the classification result to determine the rational balls B p,q which admit smooth embeddings in CP 2 induced by a genus-1 horizontal decomposition. It turns out that in this way we are able to find the largest family so far of B p,q 's smoothly embedded into the complex projective plane. The family contains all such rational balls previously known to embed into CP 2 and many more. We show that each rational ball of the family admits a symplectic embedding in CP 2 if and only if it admits an almost complex embedding.In order to state our results we need to recall the definition of a horizontal decomposition [12]. Let H be a handlebody decomposition of a smooth, oriented 4-dimensional cobordism X :+ X be the cobordism obtained by attaching the 2-handles along a framed link L = Acccording to [12], the link L and the decompos...