Dehn lemma
Let a three-dimensional manifold contain a two-dimensional cell
with self-intersections and with a simple closed polygonal curve
without singular points as boundary; then there exists a two-dimensional cell
with boundary
which can be piecewise-linearly imbedded into
. 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
be a compact three-dimensional manifold and let
be a component of its boundary; if the kernel of the homomorphism
is non-trivial, there exists a simple loop on
which is not homotopic to zero in
and is homotopic to zero in
. The loop theorem and Dehn's lemma are usually employed together. They may be combined to yield the following theorem: If
is a three-dimensional manifold with boundary
and if the kernel of the imbedding homomorphism
is non-trivial, then
contains a piecewise-linearly imbedded two-dimensional disc
, the boundary of which lies in
and which is not contractible in
. 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
is an oriented three-dimensional manifold with
, then
contains a submanifold
homeomorphic to a two-dimensional sphere and which is not homotopic to zero in
.
These results have numerous applications in the topology of three-dimensional manifolds and, in particular, in knot theory. Thus, if is a knot, then
is isomorphic to
if and only if
is a trivial knot. The following conditions are equivalent for an
-component link
in
: 1)
; 2)
is a free product of two non-trivial groups; and 3)
contains a submanifold
that is homeomorphic to a two-dimensional sphere such that both components of
contain points from
. In particular, if
is a knot (i.e.
), then
(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) |
Comments
References
[a1] | W.S. Massey, "Algebraic topology: an introduction" , Springer (1977) |
Dehn lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dehn_lemma&oldid=33393