Poincaré space

A Poincaré space of formal dimension is a topological space in which is given an element such that the homomorphism given by is an isomorphism for each (here is Whitney's product operation, the cap product). Moreover, is called the Poincaré duality isomorphism and the element generates the group . Any closed orientable -dimensional connected topological manifold is a Poincaré space of formal dimension ; is taken to be an orientation (the fundamental class) of the manifold.

Let be a finite cellular space imbedded in a Euclidean space of large dimension , let be a closed regular neighbourhood of this imbedding and let be its boundary. The standard mapping turns out to be a (Serre) fibration. : is a Poincaré space of formal dimension if and only if the fibre of this fibration is homotopy equivalent to the sphere . The described fibration which arises when 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 . Moreover, the cone of the projection is the Thom space of the normal spherical fibration over .

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

One also considers Poincaré pairs (generalizations of the concept of a manifold with boundary), where for a certain generator and any there is a Poincaré duality isomorphism: 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 -dimensional Poincaré space one means a closed -dimensional manifold with homology groups (cf. Homology group) isomorphic to the homology groups of the -dimensional sphere ; these are also called homology spheres.

A simply-connected Poincaré space is homotopy equivalent to a sphere. For a group that is realizable as the fundamental group of a certain Poincaré space one has , where are the homology groups of the group . Conversely, for any and any finitely-presented group with there exist an -dimensional Poincaré space with .

For these conditions are insufficient to realize the group in the form . 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 : , which is the fundamental group of the dodecahedral space — historically the first example of a Poincaré space.