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A Poincaré space of formal dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p0731101.png" /> is a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p0731102.png" /> in which is given an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p0731103.png" /> such that the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p0731104.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p0731105.png" /> is an isomorphism for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p0731106.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p0731107.png" /> is Whitney's product operation, the cap product). Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p0731108.png" /> is called the Poincaré duality isomorphism and the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p0731109.png" /> generates the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311010.png" />. Any closed orientable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311011.png" />-dimensional connected topological manifold is a Poincaré space of formal dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311013.png" /> is taken to be an orientation (the fundamental class) of the manifold.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311014.png" /> be a finite cellular space imbedded in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311015.png" /> of large dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311016.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311017.png" /> be a closed regular neighbourhood of this imbedding and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311018.png" /> be its boundary. The standard mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311019.png" /> turns out to be a (Serre) fibration. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311020.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311021.png" /> is a Poincaré space of formal dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311022.png" /> if and only if the fibre of this fibration is homotopy equivalent to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311023.png" />. The described fibration which arises when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311024.png" /> is a Poincaré space (the fibre of which is a sphere) is unique up to the standard equivalence and is called the normal spherical fibration, or the Spivak fibration, of the Poincaré space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311025.png" />. Moreover, the cone of the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311026.png" /> is the [[Thom space|Thom space]] of the normal spherical fibration over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311027.png" />.
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If one restricts just to homology with coefficients in a certain field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311028.png" />, then a so-called Poincaré space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311029.png" /> is obtained.
+
A Poincaré space of formal dimension  $  n $
 +
is a topological space  $  X $
 +
in which is given an element  $  \mu \in H _ {n} ( X) = \mathbf Z $
 +
such that the homomorphism  $  \cap \mu : H  ^ {k} ( X) \rightarrow H _ {n-} k ( X) $
 +
given by  $  x \rightarrow x \cap \mu $
 +
is an isomorphism for each  $  k $(
 +
here  $  \cap $
 +
is Whitney's product operation, the cap product). Moreover, $  \cap \mu $
 +
is called the Poincaré duality isomorphism and the element  $  \mu $
 +
generates the group  $  H _ {n} ( X) = \mathbf Z $.  
 +
Any closed orientable  $  n $-
 +
dimensional connected topological manifold is a Poincaré space of formal dimension  $  n $;
 +
$  \mu $
 +
is taken to be an orientation (the fundamental class) of the manifold.
  
One also considers Poincaré pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311030.png" /> (generalizations of the concept of a manifold with boundary), where for a certain generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311031.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311032.png" /> there is a Poincaré duality isomorphism:
+
Let  $  X $
 +
be a finite cellular space imbedded in a Euclidean space  $  \mathbf R  ^ {N} $
 +
of large dimension  $  N $,
 +
let  $  U $
 +
be a closed regular neighbourhood of this imbedding and let  $  \partial  U $
 +
be its boundary. The standard mapping  $  p : \partial  U \rightarrow X $
 +
turns out to be a (Serre) fibration. $  Theorem $:
 +
$  X $
 +
is a Poincaré space of formal dimension  $  n $
 +
if and only if the fibre of this fibration is homotopy equivalent to the sphere  $  S  ^ {N-} n- 1 $.  
 +
The described fibration which arises when  $  X $
 +
is a Poincaré space (the fibre of which is a sphere) is unique up to the standard equivalence and is called the normal spherical fibration, or the Spivak fibration, of the Poincaré space  $  X $.  
 +
Moreover, the cone of the projection  $  p : \partial  U \rightarrow X $
 +
is the [[Thom space|Thom space]] of the normal spherical fibration over  $  X $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311033.png" /></td> </tr></table>
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If one restricts just to homology with coefficients in a certain field  $  F $,
 +
then a so-called Poincaré space over  $  F $
 +
is obtained.
 +
 
 +
One also considers Poincaré pairs  $  ( X , A ) $(
 +
generalizations of the concept of a manifold with boundary), where for a certain generator  $  \mu \in H _ {n} ( X , A ) = \mathbf Z $
 +
and any  $  k $
 +
there is a Poincaré duality isomorphism:
 +
 
 +
$$
 +
\cap \mu : H  ^ {k} ( X)  \rightarrow  H _ {n-} k ( X , A ) .
 +
$$
  
 
Poincaré spaces naturally arise in problems on the existence and the classification of structures on manifolds. The problem of smoothing (triangulation) of a Poincaré space is also interesting, that is, to find a smooth (piecewise-linear), closed manifold that is homotopy equivalent to a given Poincaré space.
 
Poincaré spaces naturally arise in problems on the existence and the classification of structures on manifolds. The problem of smoothing (triangulation) of a Poincaré space is also interesting, that is, to find a smooth (piecewise-linear), closed manifold that is homotopy equivalent to a given Poincaré space.
  
Sometimes, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311035.png" />-dimensional Poincaré space one means a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311036.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311037.png" /> with homology groups (cf. [[Homology group|Homology group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311038.png" /> isomorphic to the homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311039.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311040.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311041.png" />; these are also called homology spheres.
+
Sometimes, by $  n $-
 +
dimensional Poincaré space one means a closed $  n $-
 +
dimensional manifold $  M $
 +
with homology groups (cf. [[Homology group|Homology group]]) $  H _ {i} ( M) $
 +
isomorphic to the homology groups $  H _ {i} ( S  ^ {n} ) $
 +
of the $  n $-
 +
dimensional sphere $  S  ^ {n} $;  
 +
these are also called homology spheres.
  
A simply-connected Poincaré space is homotopy equivalent to a sphere. For a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311042.png" /> that is realizable as the [[Fundamental group|fundamental group]] of a certain Poincaré space one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311044.png" /> are the homology groups of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311045.png" />. Conversely, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311046.png" /> and any finitely-presented group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311047.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311048.png" /> there exist an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311049.png" />-dimensional Poincaré space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311050.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311051.png" />.
+
A simply-connected Poincaré space is homotopy equivalent to a sphere. For a group $  \pi $
 +
that is realizable as the [[Fundamental group|fundamental group]] of a certain Poincaré space one has $  H _ {1} ( \pi ) = H _ {2} ( \pi ) = 0 $,  
 +
where $  H _ {i} ( \pi ) $
 +
are the homology groups of the group $  \pi $.  
 +
Conversely, for any $  n \geq  5 $
 +
and any finitely-presented group $  \pi $
 +
with $  H _ {1} ( \pi ) = H _ {2} ( \pi ) = 0 $
 +
there exist an $  n $-
 +
dimensional Poincaré space $  M $
 +
with $  \pi _ {1} ( M) = \pi $.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311052.png" /> these conditions are insufficient to realize the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311053.png" /> in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311054.png" />. So, for example, the fundamental group of any three-dimensional Poincaré space admits a presentation with the same number of generators and relations. The only finite group which is realizable as the fundamental group of a three-dimensional Poincaré space is the binary icosahedral group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311055.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073110/p07311056.png" />, which is the fundamental group of the [[Dodecahedral space|dodecahedral space]] — historically the first example of a Poincaré space.
+
For $  n = 3 , 4 $
 +
these conditions are insufficient to realize the group $  \pi $
 +
in the form $  \pi = \pi _ {1} ( M) $.  
 +
So, for example, the fundamental group of any three-dimensional Poincaré space admits a presentation with the same number of generators and relations. The only finite group which is realizable as the fundamental group of a three-dimensional Poincaré space is the binary icosahedral group < x , y $:  
 +
$  x  ^ {2} = y  ^ {5} = 1 > $,  
 +
which is the fundamental group of the [[Dodecahedral space|dodecahedral space]] — historically the first example of a Poincaré space.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.B. Browder,  "Surgery on simply connected manifolds" , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.B. Browder,  "Surgery on simply connected manifolds" , Springer  (1972)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


A Poincaré space of formal dimension $ n $ is a topological space $ X $ in which is given an element $ \mu \in H _ {n} ( X) = \mathbf Z $ such that the homomorphism $ \cap \mu : H ^ {k} ( X) \rightarrow H _ {n-} k ( X) $ given by $ x \rightarrow x \cap \mu $ is an isomorphism for each $ k $( here $ \cap $ is Whitney's product operation, the cap product). Moreover, $ \cap \mu $ is called the Poincaré duality isomorphism and the element $ \mu $ generates the group $ H _ {n} ( X) = \mathbf Z $. Any closed orientable $ n $- dimensional connected topological manifold is a Poincaré space of formal dimension $ n $; $ \mu $ is taken to be an orientation (the fundamental class) of the manifold.

Let $ X $ be a finite cellular space imbedded in a Euclidean space $ \mathbf R ^ {N} $ of large dimension $ N $, let $ U $ be a closed regular neighbourhood of this imbedding and let $ \partial U $ be its boundary. The standard mapping $ p : \partial U \rightarrow X $ turns out to be a (Serre) fibration. $ Theorem $: $ X $ is a Poincaré space of formal dimension $ n $ if and only if the fibre of this fibration is homotopy equivalent to the sphere $ S ^ {N-} n- 1 $. The described fibration which arises when $ X $ is a Poincaré space (the fibre of which is a sphere) is unique up to the standard equivalence and is called the normal spherical fibration, or the Spivak fibration, of the Poincaré space $ X $. Moreover, the cone of the projection $ p : \partial U \rightarrow X $ is the Thom space of the normal spherical fibration over $ X $.

If one restricts just to homology with coefficients in a certain field $ F $, then a so-called Poincaré space over $ F $ is obtained.

One also considers Poincaré pairs $ ( X , A ) $( generalizations of the concept of a manifold with boundary), where for a certain generator $ \mu \in H _ {n} ( X , A ) = \mathbf Z $ and any $ k $ there is a Poincaré duality isomorphism:

$$ \cap \mu : H ^ {k} ( X) \rightarrow H _ {n-} k ( X , A ) . $$

Poincaré spaces naturally arise in problems on the existence and the classification of structures on manifolds. The problem of smoothing (triangulation) of a Poincaré space is also interesting, that is, to find a smooth (piecewise-linear), closed manifold that is homotopy equivalent to a given Poincaré space.

Sometimes, by $ n $- dimensional Poincaré space one means a closed $ n $- dimensional manifold $ M $ with homology groups (cf. Homology group) $ H _ {i} ( M) $ isomorphic to the homology groups $ H _ {i} ( S ^ {n} ) $ of the $ n $- dimensional sphere $ S ^ {n} $; these are also called homology spheres.

A simply-connected Poincaré space is homotopy equivalent to a sphere. For a group $ \pi $ that is realizable as the fundamental group of a certain Poincaré space one has $ H _ {1} ( \pi ) = H _ {2} ( \pi ) = 0 $, where $ H _ {i} ( \pi ) $ are the homology groups of the group $ \pi $. Conversely, for any $ n \geq 5 $ and any finitely-presented group $ \pi $ with $ H _ {1} ( \pi ) = H _ {2} ( \pi ) = 0 $ there exist an $ n $- dimensional Poincaré space $ M $ with $ \pi _ {1} ( M) = \pi $.

For $ n = 3 , 4 $ these conditions are insufficient to realize the group $ \pi $ in the form $ \pi = \pi _ {1} ( M) $. So, for example, the fundamental group of any three-dimensional Poincaré space admits a presentation with the same number of generators and relations. The only finite group which is realizable as the fundamental group of a three-dimensional Poincaré space is the binary icosahedral group $ < x , y $: $ x ^ {2} = y ^ {5} = 1 > $, which is the fundamental group of the dodecahedral space — historically the first example of a Poincaré space.

References

[1] W.B. Browder, "Surgery on simply connected manifolds" , Springer (1972)
How to Cite This Entry:
Poincaré space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_space&oldid=23492
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article