Poincaré space

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)