Dehn lemma

Let a three-dimensional manifold $M$ contain a two-dimensional cell $D$ with self-intersections and with a simple closed polygonal curve $C$ without singular points as boundary; then there exists a two-dimensional cell $D_0$ with boundary $C$ which can be piecewise-linearly imbedded into $M$. Dehn's lemma was introduced in [1], but its proofs contained gaps; for a complete proof see [2]. The following result, known as the loop theorem, is connected with Dehn's lemma: Let $M$ be a compact three-dimensional manifold and let $N$ be a component of its boundary; if the kernel of the homomorphism $\pi_1(N)\to\pi_1(M)$ is non-trivial, there exists a simple loop on $N$ which is not homotopic to zero in $N$ and is homotopic to zero in $M$. The loop theorem and Dehn's lemma are usually employed together. They may be combined to yield the following theorem: If $M$ is a three-dimensional manifold with boundary $N$ and if the kernel of the imbedding homomorphism $\pi_1(N)\to\pi_1(M)$ is non-trivial, then $M$ contains a piecewise-linearly imbedded two-dimensional disc $D$, the boundary of which lies in $N$ and which is not contractible in $N$. A related theorem is the sphere theorem which, in conjunction with Dehn's lemma and the loop theorem, is one of the principal tools in the topology of three-dimensional manifolds: If $M$ is an oriented three-dimensional manifold with $\pi_2(M)\neq0$, then $M$ contains a submanifold $\Sigma$ homeomorphic to a two-dimensional sphere and which is not homotopic to zero in $M$.

These results have numerous applications in the topology of three-dimensional manifolds and, in particular, in knot theory. Thus, if $K$ is a knot, then $\pi_1(S^3\setminus K)$ is isomorphic to $\mathbf Z$ if and only if $K$ is a trivial knot. The following conditions are equivalent for an $n$-component link $L$ in $S^3$: 1) $\pi_2(S^3\setminus L)\neq0$; 2) $\pi_1(S^3\setminus L)$ is a free product of two non-trivial groups; and 3) $S^3\setminus L$ contains a submanifold $N$ that is homeomorphic to a two-dimensional sphere such that both components of $S^3\setminus N$ contain points from $L$. In particular, if $L$ is a knot (i.e. $n=1$), then $\pi_2(S^3\setminus L)=0$ (the theorem on asphericity of knots).

References

[1]	M. Dehn, "Ueber die Topologie des dreidimensionalen Raumes" Math. Ann. , 69 (1910) pp. 137–168
[2]	C.D. Papakyriakopoulos, "On Dehn's lemma and the asphericity of knots" Ann. of Math. , 66 (1957) pp. 1–26
[3]	J.R. Stallings, "Group theory and three-dimensional manifolds" , Yale Univ. Press (1971)
[a1]	W.S. Massey, "Algebraic topology: an introduction" , Springer (1977)