Configurations of spheres in ${\#}^n\mathbb{C}{P}^2$

Abstract

By taking the complements of embeddings of sphere plumbings in connected sums of $\mathbb{C}{P}^2$, we construct examples of simply connected four-manifolds with lens space boundary and $b_2=1$. The resulting boundaries include many lens spaces that cannot come from integer surgery on any knot in $S^3$, so the corresponding four-manifolds cannot be built by attaching a single two-handle to $B^4$. Using similar constructions, we give an example of an embedded sphere in ${\#}^4\mathbb{C}{P}^2$ with self-intersection number $20$, and conjecture that this is the maximum possible.