Cut-and-paste operations and exotic 4-manifolds

We introduce new cut-and-paste operations that can be used to construct exotic 4-manifolds. The idea is to locate a configuration of 2-spheres embedded in an ambient 4-manifold, cut out a neighborhood (called a plumbing), and glue in its place a 4-manifold with smaller homology. This serves to generalize the rational blowdown, an operation that has been successfully used in the past to construct new exotic 4-manifolds. In particular, we will develop the notion of k-replaceable plumbings and use a 2-replaceable plumbing to construct a symplectic exotic CP2#(-6CP2). Heuristically, a k-replaceable plumbing is a plumbing that can be “symplectically replaced” by a manifold with Euler characteristic k. It will turn out that 1-replaceable plumbings are precisely those that can be rationally blown down.

We then explore the possible existence of non-simply connected plumbings that are 0-replaceable. We first construct examples of plumbings that can be “smoothly replaced” by rational homology S1 × D3s and use such replacements to construct 4-manifolds that are homeomorphic to well-known 4-manifolds. For example, we will construct a 4-manifold that is homeomorphic to, but not obviously diffeomorphic to, CP2#(-CP2). This would be the first example of an exotic CP2#(-CP2). We will also build machinery that can be helpful in determining whether this operation yields exotic 4-manifolds and we will explore the possibility of this being a symplectic operation by classifying the tight contact structures with no Giroux torsion on plumbed 3-manifolds.