Simple groups and complements of smooth surfaces in simply connected $4$-manifolds

Sam Hughes; Daniel Ruberman

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arXiv · 2402.01921

Simple groups and complements of smooth surfaces in simply connected $4$-manifolds

Sam Hughes·Daniel Ruberman

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Abstract

For each integer $n$ we construct an oriented closed simply connected $4$-manifold $X$ admitting a smoothly embedded closed connected surface $\Sigma$ of self-intersection number $n$ such that the complement of the surface has non-trivial fundamental group. This answers a question of Kronheimer in Kirby's 1997 problem list. The proof combines a topological construction with homological properties of simple groups such as Thompson's group $V$ and certain sporadic finite simple groups.

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Sam Hughes, Daniel Ruberman. 2024-02-02. Simple groups and complements of smooth surfaces in simply connected $4$-manifolds. https://arxiv.org/abs/2402.01921

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