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

2024/02/02 by Sam Hughes, Hughes, Sam, Daniel Ruberman +1
Mathematics · #20E32 #20J06 (secondary) #57R40 (primary) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.GT #msc:20E32 #msc:20J06 #msc:57R40

paper · pdf · doi:10.48550/arxiv.2402.01921

11 pages, to appear in Journal of Topology

openalex publication_date 2024/02/02 · openalex created_date 2024/02/08 · arxiv created 2026/07/30 · arxiv updated 2026/07/31 · openalex updated_date 2026/08/02

Abstract

For each integer n we construct an oriented closed simply connected 4-manifold X admitting a smoothly embedded closed connected surface Σ 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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