Volume 16, issue 5 (2016)

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The simple loop conjecture for $3$–manifolds modeled on Sol

Drew Zemke

Algebraic & Geometric Topology 16 (2016) 3051–3071
Abstract

The simple loop conjecture for $3$–manifolds states that every $2$–sided immersion of a closed surface into a $3$–manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the loop theorem to immersed surfaces. We prove the conjecture in the case that the target $3$–manifold admits a geometric structure modeled on $Sol$.

Keywords
simple loop conjecture, Sol geometry
Primary: 57M35
Secondary: 57M50