The simple loop conjecture for 3–manifolds modeled on Sol

Zemke, Drew

doi:10.2140/agt.2016.16.3051

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The simple loop conjecture for

$3$ –manifolds modeled on Sol

Drew Zemke

Algebraic & Geometric Topology 16 (2016) 3051–3071

DOI: 10.2140/agt.2016.16.3051

Abstract

The simple loop conjecture for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>$ –manifolds states that every $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn></math>$ –sided immersion of a closed surface into a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>$ –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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>$ –manifold admits a geometric structure modeled on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Sol</mo></math>$ .

Keywords

simple loop conjecture, Sol geometry

Mathematical Subject Classification 2010

Primary: 57M35

Secondary: 57M50

References

Publication

Received: 30 December 2015

Revised: 14 March 2016

Accepted: 28 March 2016

Published: 7 November 2016

Authors

Drew Zemke
Department of Mathematics Cornell University 310 Malott Hall Ithaca, NY 14853 United States
http://www.math.cornell.edu/~zemke/

Volume 16, issue 5 (2016)