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

Sam Hughes, Daniel Ruberman

Abstract

For each integer $n$ we construct a simply connected $4$-manifold $X$ admitting a smoothly embedded 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.

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

Abstract

For each integer we construct a simply connected -manifold admitting a smoothly embedded surface of self intersection number 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 and certain sporadic finite simple groups.
Paper Structure (5 sections, 4 theorems, 3 equations, 1 table)

This paper contains 5 sections, 4 theorems, 3 equations, 1 table.

Table of Contents

  1. Introduction
  2. From group theory to surface complements
  3. From Seifert fibred 3-manifolds to Thompson's group
  4. Finite simple groups
  5. The non-primitive case

Key Result

Theorem A

For any non-zero $n$, there is a simply connected $4$-manifold $X$ and surface $\Sigma$ smoothly embedded in $X$ where $\Sigma \cdot \Sigma = n$ and $\pi_1(X-\Sigma) \neq \{1\}$.

Theorems & Definitions (8)

  • Theorem A
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2: Higman--Thompson groups