# Loop theorem

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In mathematics, in the [topology](/source/Topology) of [3-manifolds](/source/3-manifold), the **loop theorem** is a generalization of [Dehn's lemma](/source/Dehn's_lemma). The loop theorem was first proven by [Christos Papakyriakopoulos](/source/Christos_Papakyriakopoulos) in 1956, along with Dehn's lemma and the [Sphere theorem](/source/Sphere_theorem_(3-manifolds)).

A simple and useful version of the loop theorem states that if for some 3-dimensional manifold *M* with boundary *∂M* there is a map

- f\colon (D^2,\partial D^2)\to (M,\partial M)

with f|\partial D^2 not nullhomotopic in \partial M, then there is an embedding with the same property.

The following version of the loop theorem, due to [John Stallings](/source/John_Stallings), is given in the standard 3-manifold treatises (such as Hempel or Jaco):

Let M be a [3-manifold](/source/3-manifold) and let S be a connected surface in \partial M. Let N\subset \pi_1(S) be a [normal subgroup](/source/Normal_subgroup) such that \mathop{\mathrm{ker}}(\pi_1(S) \to \pi_1(M)) - N \neq \emptyset. Let f \colon D^2\to M be a **continuous map** such that f(\partial D^2)\subset S and [f|\partial D^2]\notin N. Then there exists an **embedding** g\colon D^2\to M such that g(\partial D^2)\subset S and [g|\partial D^2]\notin N.

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.

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)](/source/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](/source/S.-T._Yau), which also crucially relies on the tower construction.

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](/source/Friedhelm_Waldhausen) as part of his solution to the word problem for [Haken manifolds](/source/Haken_manifold); 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](/source/Haken_hierarchy). Proofs were later written up, by [Klaus Johannson](/source/Klaus_Johannson), [Marc Lackenby](/source/Marc_Lackenby), and Iain Aitchison with [Hyam Rubinstein](/source/Hyam_Rubinstein).

## Corollary

One easy corollary of the loop theorem is a following: Let M be a compact orientable irreducible 3-manifold. Then \partial M is incompressible if and only if \pi_1(F)\to\pi_1(M) is injective for each component F of \partial M.

## References

- W. Jaco, *Lectures on 3-manifolds topology*, A.M.S. regional conference series in Math 43.
- J. Hempel, *3-manifolds*, Princeton University Press 1976.
- Hatcher, *Notes on basic 3-manifold topology*, [available online](https://pi.math.cornell.edu/~hatcher/3M/3Mdownloads.html)

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Adapted from the Wikipedia article [Loop theorem](https://en.wikipedia.org/wiki/Loop_theorem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Loop_theorem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.