View source for Poincaré space

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.

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" />.

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.

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:

<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>

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.

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" />.

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.B. Browder,   "Surgery on simply connected manifolds" , Springer  (1972)</TD></TR></table>