Dehn's lemma

Dehn's lemma is a fundamental theorem from the theory of 3-dimensional manifolds in topology . It is originally based on Max Dehn back, but was only in 1957 by Christos Papakyriakopoulos demonstrated along with a more general statement, the so-called loop set (Engl. Loop Theorem ). Waldhausen gave another proof in 1968 using hierarchies in hook manifolds .

Just like the theorem of spheres , it establishes a connection between the homotopy theory (which can be formulated in algebraic terms) and the geometric topology of 3-manifolds; both theorems form the basis for large parts of the theory of 3-manifolds.

Dehn's lemma

Be a 3-manifold and a continuous mapping of the circular disc, on an area of the edge of an embedding with is. ${\displaystyle M}$ ${\displaystyle f\colon \mathbb {D} ^{2}\rightarrow M}$ ${\displaystyle U}$ ${\displaystyle \partial \mathbb {D} ^{2}}$ ${\displaystyle f^{-1}(f(A))=A}$

Then there is an embedding with . ${\displaystyle g\colon \mathbb {D} ^{2}\rightarrow M}$ ${\displaystyle f\mid _{\partial \mathbb {D} ^{2}}=g\mid _{\partial \mathbb {D} ^{2}}}$

Loop set

Let be a 3-manifold, a connected component of the boundary . ${\displaystyle M}$ ${\displaystyle F}$ ${\displaystyle \partial M}$

If is not injective, then there is an actual embedding with ${\displaystyle \pi _{1}F\rightarrow \pi _{1}M}$ ${\displaystyle g\colon (\mathbb {D} ^{2},\partial D^{2})\rightarrow (M,F)}$

${\displaystyle \left[g\mid _{\partial \mathbb {D} ^{2}}\right]\not =0\in ker(\pi _{1}F\rightarrow \pi _{1}M)}$.

More generally, if under the above conditions and is a normal divisor , then there is an actual embedding with ${\displaystyle N\subset \pi _{1}F}$ ${\displaystyle ker(\pi _{1}F\rightarrow \pi _{1}M)\setminus N\not =\emptyset }$ ${\displaystyle g\colon (\mathbb {D} ^{2},\partial D^{2})\rightarrow (M,F)}$

${\displaystyle \left[g\mid _{\partial \mathbb {D} ^{2}}\right]\not \in N}$.

Application: Incompressible surfaces

A (embedded or in the edge) in a 3-manifold actually embedded area of sex is incompressible, if not in embedded circular disk with and there. ${\displaystyle F\subset M}$ ${\displaystyle g\geq 1}$ ${\displaystyle M}$ ${\displaystyle \mathbb {D} \subset M}$ ${\displaystyle \mathbb {D} \cap F=\partial \mathbb {D} }$ ${\displaystyle \left[\partial D\right]\not =0\in \pi _{1}F}$

The following homotopy-theoretical characterization of bilateral incompressible areas by gender provides a direct application of the loop theorem . ${\displaystyle g\geq 1}$

A coherent two-sided surface of gender that is actually embedded in a 3-manifold (or embedded in the edge) is incompressible if and only if ${\displaystyle M}$ ${\displaystyle F}$ ${\displaystyle g\geq 1}$

${\displaystyle \pi _{1}F\rightarrow \pi _{1}M}$

is injective.

Application: knot theory

In knot theory, it follows from Dehn's lemma that the trivial knot can be characterized by means of the knot group , that is, the fundamental group of the knot complement.

A node is trivial if and only if ${\displaystyle K\subset S^{3}}$

${\displaystyle \pi _{1}(S^{3}\setminus K)\cong \mathbb {Z} }$

applies.

literature

• Hempel, John: 3-manifolds. Reprint of the 1976 original. AMS Chelsea Publishing, Providence, RI, 2004. ISBN 0-8218-3695-1
• Jaco, William: Lectures on three-manifold topology. CBMS Regional Conference Series in Mathematics, 43rd American Mathematical Society, Providence, RI, 1980. ISBN 0-8218-1693-4
• Papakyriakopoulos, CD: On Dehn's lemma and the asphericity of knots. Ann. of Math. (2) 66: 1-26 (1957).
• Waldhausen, F .: The word problem in fundamental groups of sufficiently large irreducible 3-manifolds. Ann. of Math. (2) 88: 272-280 (1968).
• Stallings, John: Group theory and three-dimensional manifolds. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale Mathematical Monographs, 4th Yale University Press, New Haven, Conn.-London, 1971.

Web links

Hatcher: Notes on Basic 3-Manifold Topology (PDF; 665 kB)