# 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.

How to Cite This Entry:
Poincaré space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_space&oldid=48208
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article