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A [[Topological space|topological space]] each point of which has a neighbourhood homeomorphic to three-dimensional real space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t0927301.png" /> or to the closed half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t0927302.png" />. This definition is usually supplemented by the requirement that a three-dimensional manifold as a topological space be Hausdorff and have a countable base. The boundary of a three-dimensional manifold, that is, its set of points that only have neighbourhoods of the second, rather than the first, of the above types, is a [[Two-dimensional manifold|two-dimensional manifold]] without boundary. Methods of the topology of three-dimensional manifolds are very specific and therefore occupy a special place in the [[Topology of manifolds|topology of manifolds]].
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A [[Topological space|topological space]] each point of which has a neighbourhood homeomorphic to three-dimensional real space $  \mathbf R  ^ {3} $
 +
or to the closed half-space $  \mathbf R _ {+}  ^ {3} $.  
 +
This definition is usually supplemented by the requirement that a three-dimensional manifold as a topological space be Hausdorff and have a countable base. The boundary of a three-dimensional manifold, that is, its set of points that only have neighbourhoods of the second, rather than the first, of the above types, is a [[Two-dimensional manifold|two-dimensional manifold]] without boundary. Methods of the topology of three-dimensional manifolds are very specific and therefore occupy a special place in the [[Topology of manifolds|topology of manifolds]].
  
 
Examples. Some properties of three-dimensional manifolds that, in general, do not hold in higher dimensions are: an orientable three-dimensional manifold is always parallelizable; a closed three-dimensional manifold bounds some four-dimensional manifold; one can always introduce into a three-dimensional manifold piecewise-linear and differentiable structures, and any homeomorphism between two three-dimensional manifolds can be approximated by a piecewise-linear homeomorphism as well as by a diffeomorphism.
 
Examples. Some properties of three-dimensional manifolds that, in general, do not hold in higher dimensions are: an orientable three-dimensional manifold is always parallelizable; a closed three-dimensional manifold bounds some four-dimensional manifold; one can always introduce into a three-dimensional manifold piecewise-linear and differentiable structures, and any homeomorphism between two three-dimensional manifolds can be approximated by a piecewise-linear homeomorphism as well as by a diffeomorphism.
  
One of the most widespread methods of describing a three-dimensional manifold is the use of Heegaard decompositions and the Heegaard diagrams closely related to them (cf. [[Heegaard decomposition|Heegaard decomposition]]; [[Heegaard diagram|Heegaard diagram]]). The essence of this method is that any closed oriented three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t0927303.png" /> can be decomposed into two submanifolds with a common boundary, each of which is homeomorphic to a standard complete pretzel (or handlebody, cf. [[Handle theory|Handle theory]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t0927304.png" /> of some genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t0927305.png" />. In other words, a three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t0927306.png" /> can be obtained by glueing two copies of a complete pretzel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t0927307.png" /> along their boundaries by some homeomorphism. This fact enables one to reduce many problems in the topology of three-dimensional manifolds to those in the topology of surfaces. The smallest possible number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t0927308.png" /> is called the genus of the three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t0927309.png" />. Another useful method of describing a three-dimensional manifold is based on the existence of a close connection between three-dimensional manifolds and links in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273010.png" /> (cf. [[Knot theory|Knot theory]]): Any closed oriented three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273011.png" /> can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273012.png" />, where the four-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273013.png" /> is obtained from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273014.png" />-ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273015.png" /> by attaching handles of index 2 along the components of some framed link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273017.png" />. Equivalently, a three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273018.png" /> can be obtained from the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273019.png" /> by spherical [[Surgery|surgery]]. It may be required in addition that all the components of the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273020.png" /> have even framings, and then the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273021.png" /> thus obtained is parallelizable. Often one uses the representation of a three-dimensional manifold as the space of a ramified covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273022.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273023.png" /> is a link in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273024.png" />, then any finitely-sheeted covering space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273025.png" /> can be compactified by certain circles to give a closed three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273026.png" />. The natural projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273027.png" />, which is locally homeomorphic outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273028.png" />, is called the ramified covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273029.png" /> with ramification along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273030.png" />. Any three-dimensional manifold of genus 2 is a double covering of the sphere with ramification along some link, while in the case of a three-dimensional manifold of arbitrary genus one can only guarantee the existence of a triple covering with ramification along some knot. This circumstance is the main cause why the three-dimensional Poincaré conjecture and the problem of the algorithmic recognition of a sphere have so far (1984) only been solved in the class of three-dimensional manifolds of genus 2.
+
One of the most widespread methods of describing a three-dimensional manifold is the use of Heegaard decompositions and the Heegaard diagrams closely related to them (cf. [[Heegaard decomposition|Heegaard decomposition]]; [[Heegaard diagram|Heegaard diagram]]). The essence of this method is that any closed oriented three-dimensional manifold $  M $
 +
can be decomposed into two submanifolds with a common boundary, each of which is homeomorphic to a standard complete pretzel (or handlebody, cf. [[Handle theory|Handle theory]]) $  V $
 +
of some genus $  n $.  
 +
In other words, a three-dimensional manifold $  M $
 +
can be obtained by glueing two copies of a complete pretzel $  V $
 +
along their boundaries by some homeomorphism. This fact enables one to reduce many problems in the topology of three-dimensional manifolds to those in the topology of surfaces. The smallest possible number $  n $
 +
is called the genus of the three-dimensional manifold $  M $.  
 +
Another useful method of describing a three-dimensional manifold is based on the existence of a close connection between three-dimensional manifolds and links in $  S  ^ {3} $(
 +
cf. [[Knot theory|Knot theory]]): Any closed oriented three-dimensional manifold $  M $
 +
can be represented in the form $  M = \partial  W $,  
 +
where the four-dimensional manifold $  W $
 +
is obtained from the $  4 $-
 +
ball $  B  ^ {4} $
 +
by attaching handles of index 2 along the components of some framed link $  L $
 +
in $  S  ^ {3} = \partial  B  ^ {4} $.  
 +
Equivalently, a three-dimensional manifold $  M $
 +
can be obtained from the sphere $  S  ^ {3} $
 +
by spherical [[Surgery|surgery]]. It may be required in addition that all the components of the link $  L $
 +
have even framings, and then the manifold $  W $
 +
thus obtained is parallelizable. Often one uses the representation of a three-dimensional manifold as the space of a ramified covering of $  S  ^ {3} $.  
 +
If $  L $
 +
is a link in $  S  ^ {3} $,  
 +
then any finitely-sheeted covering space of $  S _ {3} /L $
 +
can be compactified by certain circles to give a closed three-dimensional manifold $  M $.  
 +
The natural projection $  p: M \rightarrow S  ^ {3} $,  
 +
which is locally homeomorphic outside $  p  ^ {-} 1 ( L) $,  
 +
is called the ramified covering of $  S  ^ {3} $
 +
with ramification along $  L $.  
 +
Any three-dimensional manifold of genus 2 is a double covering of the sphere with ramification along some link, while in the case of a three-dimensional manifold of arbitrary genus one can only guarantee the existence of a triple covering with ramification along some knot. This circumstance is the main cause why the three-dimensional Poincaré conjecture and the problem of the algorithmic recognition of a sphere have so far (1984) only been solved in the class of three-dimensional manifolds of genus 2.
  
The main problem in the topology of three-dimensional manifolds is that of their classification. A three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273031.png" /> is said to be simple if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273032.png" /> implies that exactly one of the manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273034.png" /> is a sphere. Every compact three-dimensional manifold decomposes into a connected sum of a finite number of simple three-dimensional manifolds. This decomposition is unique in the orientable case and is unique up to replacement of the direct product by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273035.png" /> in the non-orientable case. Instead of the notion of a simple three-dimensional manifold, it is often more useful to use the notion of an irreducible three-dimensional manifold, that is, a manifold in which every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273036.png" />-sphere bounds a ball. The class of irreducible three-dimensional manifolds differs from that of simple three-dimensional manifolds by just three manifolds: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273039.png" />. Here the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273040.png" /> is irreducible, but is usually not considered to be simple, while the manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273042.png" /> are simple but not irreducible. Irreducible three-dimensional manifolds with boundary have been fairly well studied. For example, any homotopy equivalence of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273045.png" /> are compact oriented irreducible three-dimensional manifolds with boundary, can be deformed into a homeomorphism. In the closed case it suffices for this that in addition the three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273046.png" /> is sufficiently large, i.e. that it contains some two-sided incompressible surface. Here, a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273048.png" />, is said to be incompressible if the group homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273049.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092730/t09273050.png" /> induced by the imbedding is injective. If the first homology group of a compact irreducible three-dimensional manifold is infinite, then such a surface always exists. Any compact oriented irreducible sufficiently large three-dimensional manifold whose fundamental group contains an infinite cyclic normal subgroup is a [[Seifert manifold|Seifert manifold]].
+
The main problem in the topology of three-dimensional manifolds is that of their classification. A three-dimensional manifold $  M $
 +
is said to be simple if $  M = M _ {1} \# M _ {2} $
 +
implies that exactly one of the manifolds $  M _ {1} $,  
 +
$  M _ {2} $
 +
is a sphere. Every compact three-dimensional manifold decomposes into a connected sum of a finite number of simple three-dimensional manifolds. This decomposition is unique in the orientable case and is unique up to replacement of the direct product by $  S  ^ {2} \widetilde \times  S  ^ {1} $
 +
in the non-orientable case. Instead of the notion of a simple three-dimensional manifold, it is often more useful to use the notion of an irreducible three-dimensional manifold, that is, a manifold in which every $  2 $-
 +
sphere bounds a ball. The class of irreducible three-dimensional manifolds differs from that of simple three-dimensional manifolds by just three manifolds: $  S  ^ {3} $,  
 +
$  S  ^ {2} \times S  ^ {1} $
 +
and $  S  ^ {2} \widetilde \times  S  ^ {1} $.  
 +
Here the manifold $  S  ^ {3} $
 +
is irreducible, but is usually not considered to be simple, while the manifolds $  S  ^ {2} \times S  ^ {1} $
 +
and $  S  ^ {2} \widetilde \times  S  ^ {1} $
 +
are simple but not irreducible. Irreducible three-dimensional manifolds with boundary have been fairly well studied. For example, any homotopy equivalence of pairs $  f: ( M, \partial  M) \rightarrow ( M, \partial  N) $,  
 +
where $  M $,  
 +
$  N $
 +
are compact oriented irreducible three-dimensional manifolds with boundary, can be deformed into a homeomorphism. In the closed case it suffices for this that in addition the three-dimensional manifold $  M $
 +
is sufficiently large, i.e. that it contains some two-sided incompressible surface. Here, a surface $  F \subset  M $,  
 +
$  F \neq S  ^ {2} $,  
 +
is said to be incompressible if the group homomorphism from $  \pi _ {1} ( F  ) $
 +
into $  \pi _ {1} ( M) $
 +
induced by the imbedding is injective. If the first homology group of a compact irreducible three-dimensional manifold is infinite, then such a surface always exists. Any compact oriented irreducible sufficiently large three-dimensional manifold whose fundamental group contains an infinite cyclic normal subgroup is a [[Seifert manifold|Seifert manifold]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hempel,  "3-manifolds" , Princeton Univ. Press  (1976)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Waldhausen,  "On irreducible 3-manifolds which are sufficiently large"  ''Ann. of Math.'' , '''87'''  (1968)  pp. 56–88</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.H. Jaco,  "Lectures on three-manifold topology" , Amer. Math. Soc.  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hempel,  "3-manifolds" , Princeton Univ. Press  (1976)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Waldhausen,  "On irreducible 3-manifolds which are sufficiently large"  ''Ann. of Math.'' , '''87'''  (1968)  pp. 56–88</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.H. Jaco,  "Lectures on three-manifold topology" , Amer. Math. Soc.  (1980)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.E. Moise,  "Geometric topology in dimensions 2 and 3" , Springer  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.E. Moise,  "Geometric topology in dimensions 2 and 3" , Springer  (1977)</TD></TR></table>

Revision as of 08:25, 6 June 2020


A topological space each point of which has a neighbourhood homeomorphic to three-dimensional real space $ \mathbf R ^ {3} $ or to the closed half-space $ \mathbf R _ {+} ^ {3} $. This definition is usually supplemented by the requirement that a three-dimensional manifold as a topological space be Hausdorff and have a countable base. The boundary of a three-dimensional manifold, that is, its set of points that only have neighbourhoods of the second, rather than the first, of the above types, is a two-dimensional manifold without boundary. Methods of the topology of three-dimensional manifolds are very specific and therefore occupy a special place in the topology of manifolds.

Examples. Some properties of three-dimensional manifolds that, in general, do not hold in higher dimensions are: an orientable three-dimensional manifold is always parallelizable; a closed three-dimensional manifold bounds some four-dimensional manifold; one can always introduce into a three-dimensional manifold piecewise-linear and differentiable structures, and any homeomorphism between two three-dimensional manifolds can be approximated by a piecewise-linear homeomorphism as well as by a diffeomorphism.

One of the most widespread methods of describing a three-dimensional manifold is the use of Heegaard decompositions and the Heegaard diagrams closely related to them (cf. Heegaard decomposition; Heegaard diagram). The essence of this method is that any closed oriented three-dimensional manifold $ M $ can be decomposed into two submanifolds with a common boundary, each of which is homeomorphic to a standard complete pretzel (or handlebody, cf. Handle theory) $ V $ of some genus $ n $. In other words, a three-dimensional manifold $ M $ can be obtained by glueing two copies of a complete pretzel $ V $ along their boundaries by some homeomorphism. This fact enables one to reduce many problems in the topology of three-dimensional manifolds to those in the topology of surfaces. The smallest possible number $ n $ is called the genus of the three-dimensional manifold $ M $. Another useful method of describing a three-dimensional manifold is based on the existence of a close connection between three-dimensional manifolds and links in $ S ^ {3} $( cf. Knot theory): Any closed oriented three-dimensional manifold $ M $ can be represented in the form $ M = \partial W $, where the four-dimensional manifold $ W $ is obtained from the $ 4 $- ball $ B ^ {4} $ by attaching handles of index 2 along the components of some framed link $ L $ in $ S ^ {3} = \partial B ^ {4} $. Equivalently, a three-dimensional manifold $ M $ can be obtained from the sphere $ S ^ {3} $ by spherical surgery. It may be required in addition that all the components of the link $ L $ have even framings, and then the manifold $ W $ thus obtained is parallelizable. Often one uses the representation of a three-dimensional manifold as the space of a ramified covering of $ S ^ {3} $. If $ L $ is a link in $ S ^ {3} $, then any finitely-sheeted covering space of $ S _ {3} /L $ can be compactified by certain circles to give a closed three-dimensional manifold $ M $. The natural projection $ p: M \rightarrow S ^ {3} $, which is locally homeomorphic outside $ p ^ {-} 1 ( L) $, is called the ramified covering of $ S ^ {3} $ with ramification along $ L $. Any three-dimensional manifold of genus 2 is a double covering of the sphere with ramification along some link, while in the case of a three-dimensional manifold of arbitrary genus one can only guarantee the existence of a triple covering with ramification along some knot. This circumstance is the main cause why the three-dimensional Poincaré conjecture and the problem of the algorithmic recognition of a sphere have so far (1984) only been solved in the class of three-dimensional manifolds of genus 2.

The main problem in the topology of three-dimensional manifolds is that of their classification. A three-dimensional manifold $ M $ is said to be simple if $ M = M _ {1} \# M _ {2} $ implies that exactly one of the manifolds $ M _ {1} $, $ M _ {2} $ is a sphere. Every compact three-dimensional manifold decomposes into a connected sum of a finite number of simple three-dimensional manifolds. This decomposition is unique in the orientable case and is unique up to replacement of the direct product by $ S ^ {2} \widetilde \times S ^ {1} $ in the non-orientable case. Instead of the notion of a simple three-dimensional manifold, it is often more useful to use the notion of an irreducible three-dimensional manifold, that is, a manifold in which every $ 2 $- sphere bounds a ball. The class of irreducible three-dimensional manifolds differs from that of simple three-dimensional manifolds by just three manifolds: $ S ^ {3} $, $ S ^ {2} \times S ^ {1} $ and $ S ^ {2} \widetilde \times S ^ {1} $. Here the manifold $ S ^ {3} $ is irreducible, but is usually not considered to be simple, while the manifolds $ S ^ {2} \times S ^ {1} $ and $ S ^ {2} \widetilde \times S ^ {1} $ are simple but not irreducible. Irreducible three-dimensional manifolds with boundary have been fairly well studied. For example, any homotopy equivalence of pairs $ f: ( M, \partial M) \rightarrow ( M, \partial N) $, where $ M $, $ N $ are compact oriented irreducible three-dimensional manifolds with boundary, can be deformed into a homeomorphism. In the closed case it suffices for this that in addition the three-dimensional manifold $ M $ is sufficiently large, i.e. that it contains some two-sided incompressible surface. Here, a surface $ F \subset M $, $ F \neq S ^ {2} $, is said to be incompressible if the group homomorphism from $ \pi _ {1} ( F ) $ into $ \pi _ {1} ( M) $ induced by the imbedding is injective. If the first homology group of a compact irreducible three-dimensional manifold is infinite, then such a surface always exists. Any compact oriented irreducible sufficiently large three-dimensional manifold whose fundamental group contains an infinite cyclic normal subgroup is a Seifert manifold.

References

[1] J. Hempel, "3-manifolds" , Princeton Univ. Press (1976)
[2] F. Waldhausen, "On irreducible 3-manifolds which are sufficiently large" Ann. of Math. , 87 (1968) pp. 56–88
[3] W.H. Jaco, "Lectures on three-manifold topology" , Amer. Math. Soc. (1980)

Comments

References

[a1] E.E. Moise, "Geometric topology in dimensions 2 and 3" , Springer (1977)
How to Cite This Entry:
Three-dimensional manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Three-dimensional_manifold&oldid=48974
This article was adapted from an original article by S.V. Matveev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article