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Let a three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d0309301.png" /> contain a two-dimensional cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d0309302.png" /> with self-intersections and with a simple closed polygonal curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d0309303.png" /> without singular points as boundary; then there exists a two-dimensional cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d0309304.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d0309305.png" /> which can be piecewise-linearly imbedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d0309306.png" />. Dehn's lemma was introduced in [[#References|[1]]], but its proofs contained gaps; for a complete proof see [[#References|[2]]]. The following result, known as the loop theorem, is connected with Dehn's lemma: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d0309307.png" /> be a compact three-dimensional manifold and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d0309308.png" /> be a component of its boundary; if the kernel of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d0309309.png" /> is non-trivial, there exists a simple loop on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093010.png" /> which is not homotopic to zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093011.png" /> and is homotopic to zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093012.png" />. The loop theorem and Dehn's lemma are usually employed together. They may be combined to yield the following theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093013.png" /> is a three-dimensional manifold with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093014.png" /> and if the kernel of the imbedding homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093015.png" /> is non-trivial, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093016.png" /> contains a piecewise-linearly imbedded two-dimensional disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093017.png" />, the boundary of which lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093018.png" /> and which is not contractible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093019.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093020.png" /> is an oriented three-dimensional manifold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093022.png" /> contains a submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093023.png" /> homeomorphic to a two-dimensional sphere and which is not homotopic to zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093024.png" />.
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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 [[#References|[1]]], but its proofs contained gaps; for a complete proof see [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093025.png" /> is a knot, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093026.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093027.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093028.png" /> is a trivial knot. The following conditions are equivalent for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093029.png" />-component link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093031.png" />: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093032.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093033.png" /> is a free product of two non-trivial groups; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093034.png" /> contains a submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093035.png" /> that is homeomorphic to a two-dimensional sphere such that both components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093036.png" /> contain points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093037.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093038.png" /> is a knot (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093039.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030930/d03093040.png" /> (the theorem on asphericity of knots).
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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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Dehn,  "Ueber die Topologie des dreidimensionalen Raumes"  ''Math. Ann.'' , '''69'''  (1910)  pp. 137–168</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.D. Papakyriakopoulos,  "On Dehn's lemma and the asphericity of knots"  ''Ann. of Math.'' , '''66'''  (1957)  pp. 1–26</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.R. Stallings,  "Group theory and three-dimensional manifolds" , Yale Univ. Press  (1971)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  M. Dehn,  "Ueber die Topologie des dreidimensionalen Raumes"  ''Math. Ann.'' , '''69'''  (1910)  pp. 137–168</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.D. Papakyriakopoulos,  "On Dehn's lemma and the asphericity of knots"  ''Ann. of Math.'' , '''66'''  (1957)  pp. 1–26</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.R. Stallings,  "Group theory and three-dimensional manifolds" , Yale Univ. Press  (1971)</TD></TR>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W.S. Massey,  "Algebraic topology: an introduction" , Springer  (1977)</TD></TR>
 
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</table>
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.S. Massey,  "Algebraic topology: an introduction" , Springer  (1977)</TD></TR></table>
 

Latest revision as of 08:36, 1 May 2023

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)
How to Cite This Entry:
Dehn lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dehn_lemma&oldid=17859
This article was adapted from an original article by M.I. VoitsekhovskiiM.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article