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

TL;DR

This paper constructs new simply connected four-manifolds with lens space boundaries and explores sphere configurations in connected sums of complex projective planes, revealing boundaries not obtainable via knot surgery.

Contribution

It introduces novel constructions of four-manifolds with specific boundary types and sphere configurations, expanding understanding of four-manifold topology.

Findings

Constructed simply connected four-manifolds with lens space boundaries and $b_2=1$

Identified boundaries not realizable by knot surgery in $S^3$

Presented an embedded sphere with self-intersection 20 in $ abla^4 ext{CP}^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.