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| In mathematics, in the [[topology]] of [[3-manifold]]s, the '''loop theorem''' is a generalization of [[Dehn's lemma]]. The loop theorem was first proven by [[Christos Papakyriakopoulos]] in 1956, along with Dehn's lemma and the [[Sphere theorem %283-manifolds%29|Sphere theorem]].
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| A simple and useful version of the loop theorem states that if there is a map
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| :<math>f\colon (D^2,\partial D^2)\to (M,\partial M) \, </math>
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| with <math>f|\partial D^2</math> not nullhomotopic in <math>\partial M</math>, then there is an embedding with the same property.
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| The following version of the loop theorem, due to [[John Stallings]], is given in the standard 3-manifold treatises (such as Hempel or Jaco):
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| Let <math>M</math> be a [[3-manifold]] and let <math>S</math>
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| be a connected surface in <math>\partial M </math>. Let <math>N\subset
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| \pi_1(S)</math> be a [[normal subgroup]] such that <math>\mathop{\mathrm{ker}}(\pi_1(S) \to \pi_1(M)) - N \neq \emptyset</math>.
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| Let
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| :<math>f \colon D^2\to M \, </math> | |
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| be a '''continuous map''' such that
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| :<math>f(\partial D^2)\subset S \, </math>
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| and
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| :<math>[f|\partial D^2]\notin N. \, </math>
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| Then there exists an '''embedding'''
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| :<math>g\colon D^2\to M \, </math>
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| such that
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| :<math>g(\partial D^2)\subset S \, </math>
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| and
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| :<math>[g|\partial D^2]\notin N. \, </math>
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| Furthermore if one starts with a map ''f'' in general position, then for any neighborhood U of the singularity set of ''f'', we can find such a ''g'' with image lying inside the union of image of ''f'' and U.
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| Stalling's proof utilizes an adaptation, due to Whitehead and Shapiro, of Papakyriakopoulos' "tower construction". The "tower" refers to a special sequence of coverings designed to simplify lifts of the given map. The same tower construction was used by Papakyriakopoulos to prove the [[sphere theorem (3-manifolds)]], which states that a nontrivial map of a sphere into a 3-manifold implies the existence of a nontrivial ''embedding'' of a sphere. There is also a version of Dehn's lemma for minimal discs due to Meeks and S.-T. Yau, which also crucially relies on the tower construction.
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| A proof not utilizing the tower construction exists of the first version of the loop theorem. This was essentially done 30 years ago by [[Friedhelm Waldhausen]] as part of his solution to the word problem for [[Haken manifold]]s; although he recognized this gave a proof of the loop theorem, he did not write up a detailed proof. The essential ingredient of this proof is the concept of [[Haken hierarchy]]. Proofs were later written up, by [[Klaus Johannson]], Marc Lackenby, and Iain Aitchison with [[Hyam Rubinstein]].
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| ==References==
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| *W. Jaco, ''Lectures on 3-manifolds topology'', A.M.S. regional conference series in Math 43.
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| *J. Hempel, ''3-manifolds'', Princeton University Press 1976.
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| * Hatcher, ''Notes on basic 3-manifold topology'', [http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html available online]
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| [[Category:Geometric topology]]
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| [[Category:3-manifolds]]
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| [[Category:Continuous mappings]]
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| [[Category:Theorems in topology]]
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Hello. Allow me introduce the writer. Her name is Refugia Shryock. Years in the past he moved to North Dakota and his family loves it. To collect badges is what her family and her enjoy. My day occupation is a meter reader.
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