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.
Then there is an embedding with .
Loop set
Let be a 3-manifold, a connected component of the boundary .
If is not injective, then there is an actual embedding with
- .
More generally, if under the above conditions and is a normal divisor , then there is an actual embedding with
- .
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.
The following homotopy-theoretical characterization of bilateral incompressible areas by gender provides a direct application of the loop theorem .
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
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
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)