Difference between revisions of "Poincaré space"
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− | + | 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|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. | 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 | + | 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 | + | 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 | + | 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) |
Poincaré space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_space&oldid=14620