Math 2.0

Manifolds and cell complexes Three-manifolds Theorem

Dehn's lemma

Builds on 3-manifoldKnotKnot group

Statement

Let MM be a 3-manifold and let DD be the closed 22-disk. Let f ⁣:D→Mf \colon D \to M be a piecewise linear map. Its singular set is the closure of the set of points x∈Dx \in D such that f−1(f(x))f^{-1}(f(x)) contains a point other than xx. If the singular set of ff does not meet ∂D\partial D, then there is a piecewise linear embedding g ⁣:D→Mg \colon D \to M with g=fg = f on ∂D\partial D.

Remarks

Dehn published a proof in 1910, and Kneser found a gap in it. Papakyriakopoulos gave a proof in 1957 with his tower construction. One lifts ff through a tower of double coverings of regular neighborhoods of its image, until the singularities can be removed at the top. Then one pushes the embedded disk back down the tower. A key application is to knots: a tame knot KK in S3S^3 is trivial if and only if its knot group is infinite cyclic. If the knot group is ℤ\Z, the longitude of KK is null-homotopic in the complement, and Dehn's lemma turns a singular disk bounded by it into an embedded disk.

Going deeper

For a student

Dehn's lemma, the loop theorem and the sphere theorem form the classical toolkit of 33-manifold topology. They all say that a map that is nontrivial up to homotopy can be replaced by an embedding. They are used to show that an incompressible surface is the same as a two-sided surface whose inclusion is injective on fundamental groups.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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  1. 3-manifold concept
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  4. Dehn's lemma statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dehn%27s_lemma
About this page
TagFH5WSPD79F
URIm2://FH5WSPD79F
Kindstatement, theorem
WritingW0 stub
MSC57K30, 57N35
StatusThis copy of the site has no build data. Run m2 status FH5WSPD79F.
About3-manifold
Uses3-manifold, Knot, Knot group
Same as
Changed2 November 2026

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