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

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